2019
2019
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Paper 1, Section I, E
2019 commentState the Bolzano-Weierstrass theorem.
Let be a sequence of non-zero real numbers. Which of the following conditions is sufficient to ensure that converges? Give a proof or counter-example as appropriate.
(i) for some real number .
(ii) for some non-zero real number .
(iii) has no convergent subsequence.
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Paper 1, Section I, F
2019 commentLet be a real power series that diverges for at least one value of . Show that there exists a non-negative real number such that converges absolutely whenever and diverges whenever .
Find, with justification, such a number for each of the following real power series:
(i) ;
(ii) .
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Paper 1, Section II, D
2019 commentLet be a function that is continuous at at least one point . Suppose further that satisfies
for all . Show that is continuous on .
Show that there exists a constant such that for all .
Suppose that is a continuous function defined on and that satisfies the equation
for all . Show that is either identically zero or everywhere positive. What is the general form for ?
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Paper 1, Section II, D
2019 commentState and prove the Intermediate Value Theorem.
State the Mean Value Theorem.
Suppose that the function is differentiable everywhere in some open interval containing , and that . By considering the functions and defined by
and
or otherwise, show that there is a subinterval such that
Deduce that there exists with .
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Paper 1, Section II, E
2019 commentLet and be sequences of positive real numbers. Let .
(a) Show that if and converge then so does .
(b) Show that if converges then converges. Is the converse true?
(c) Show that if diverges then diverges. Is the converse true?
For part (c), it may help to show that for any there exist with
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Paper 1, Section II, F
2019 commentLet be a bounded function. Define the upper and lower integrals of . What does it mean to say that is Riemann integrable? If is Riemann integrable, what is the Riemann integral ?
Which of the following functions are Riemann integrable? For those that are Riemann integrable, find . Justify your answers.
(i)
(ii) ,
where has a base-3 expansion containing a 1;
[Hint: You may find it helpful to note, for example, that as one of the base-3 expansions of is
(iii) ,
where has a base expansion containing infinitely many .
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Paper 2, Section I, C
2019 commentThe function satisfies the inhomogeneous second-order linear differential equation
Find the solution that satisfies the conditions that and is bounded as .
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Paper 2, Section I,
2019 commentConsider the first order system
to be solved for , where the matrix and are all independent of time. Show that if is not an eigenvalue of then there is a solution of the form , with constant.
For , given
find the general solution to (1).
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Paper 2, Section II, C
2019 commentTwo cups of tea at temperatures and cool in a room at ambient constant temperature . Initially .
Cup 1 has cool milk added instantaneously at and then hot water added at a constant rate after which is modelled as follows
whereas cup 2 is left undisturbed and evolves as follows
where and are the Dirac delta and Heaviside functions respectively, and is a positive constant.
(a) Derive expressions for when and for when .
(b) Show for that
(c) Derive an expression for for .
(d) At what time is ?
(e) Find how behaves for and explain your result.
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Paper 2, Section II,
2019 commentConsider the problem of solving
subject to the initial conditions using a discrete approach where is computed at discrete times, where and
(a) By using Taylor expansions around , derive the centred-difference formula
where the value of should be found.
(b) Find the general solution of and show that this is the discrete version of the corresponding general solution to .
(c) The fully discretized version of the differential equation (1) is
By finding a particular solution first, write down the general solution to the difference equation (2). For the solution which satisfies the discretized initial conditions and , find the error in in terms of only.
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Paper 2, Section II,
2019 commentFind all power series solutions of the form to the equation
for a real constant. [It is sufficient to give a recurrence relationship between coefficients.]
Impose the condition and determine those values of for which your power series gives polynomial solutions (i.e., for sufficiently large). Give the values of for which the corresponding polynomials have degree less than 6 , and compute these polynomials. Hence, or otherwise, find a polynomial solution of
satisfying .
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Paper 2, Section II, C
2019 commentConsider the nonlinear system
(a) Show that is a constant of the motion.
(b) Find all the critical points of the system and analyse their stability. Sketch the phase portrait including the special contours with value .
(c) Find an explicit expression for in the solution which satisfies at . At what time does it reach the point
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Paper 4, Section I, A
2019 commentA rocket of mass moving at speed and ejecting fuel behind it at a constant speed relative to the rocket, is subject to an external force . Considering a small time interval , derive the rocket equation
In deep space where , how much faster does the rocket go if it burns half of its mass in fuel?
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Paper 4, Section I, A
2019 commentGalileo releases a cannonball of mass from the top of the leaning tower of Pisa, a vertical height above the ground. Ignoring the rotation of the Earth but assuming that the cannonball experiences a quadratic drag force whose magnitude is (where is the speed of the cannonball), find the time for it to hit the ground in terms of and , the acceleration due to gravity. [You may assume that is constant.]
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Paper 4, Section II, A
2019 commentIn an alien invasion, a flying saucer hovers at a fixed point , a height far above the White House, which is at point . A wrecking ball of mass is attached to one end of a light inextensible rod, also of length . The other end of the rod is attached to the flying saucer. The wrecking ball is initially at rest at point , and the angle is . At , the acceleration due to gravity is . Assume that the rotation of the Earth can be neglected and that the only force acting is Earth's gravity.
(a) Under the approximations that gravity acts everywhere parallel to the line and that the acceleration due to Earth's gravity is constant throughout the space through which the wrecking ball is travelling, find the speed with which the wrecking ball hits the White House, in terms of the constants introduced above.
(b) Taking into account the fact that gravity is non-uniform and acts toward the centre of the Earth, find the speed with which the wrecking ball hits the White House in terms of the constants introduced above and , where is the radius of the Earth, which you may assume is exactly spherical.
(c) Finally, show that
where and are constants, which you should determine.
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Paper 4, Section II, A
2019 comment(a) A particle of mass and positive charge moves with velocity in a region in which the magnetic field is constant and no other forces act, where . Initially, the particle is at position and . Write the equation of motion of the particle and then solve it to find as a function of time . Sketch its path in .
(b) For , three point particles, each of charge , are fixed at , and , respectively. Another point particle of mass and charge is constrained to move in the plane and suffers Coulomb repulsion from each fixed charge. Neglecting any magnetic fields,
(i) Find the position of an equilibrium point.
(ii) By finding the form of the electric potential near this point, deduce that the equilibrium is stable.
(iii) Consider small displacements of the point particle from the equilibrium point. By resolving forces in the directions and , show that the frequency of oscillation is
where is a constant which you should find.
[You may assume that if two identical charges are separated by a distance then the repulsive Coulomb force experienced by each of the charges is , where is a constant.]
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Paper 4, Section II, A
2019 comment(a) Writing a mass dimension as , a time dimension as , a length dimension as and a charge dimension as , write, using relations that you know, the dimensions of:
(i) force
(ii) electric field
(b) In the Large Hadron Collider at CERN, a proton of rest mass and charge is accelerated by a constant electric field . At time , the particle is at rest at the origin.
Writing the proton's position as and including relativistic effects, calculate . Use your answers to part (a) to check that the dimensions in your expression are correct.
Sketch a graph of versus , commenting on the limit.
Calculate as an explicit function of and find the non-relativistic limit at small times . What kind of motion is this?
(c) At a later time , an observer in the laboratory frame sees a cosmic microwave photon of energy hit the accelerated proton, leaving only a particle of mass in the final state. In its rest frame, the takes a time to decay. How long does it take to decay in the laboratory frame as a function of and , the speed of light in a vacuum?
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Paper 4, Section II, A
2019 commentAn inertial frame and another reference frame have a common origin , and rotates with angular velocity vector with respect to . Derive the results (a) and (b) below, where dot denotes a derivative with respect to time :
(a) The rates of change of an arbitrary vector in frames and are related by
(b) The accelerations in and are related by
where is the position vector relative to .
Just after passing the South Pole, a ski-doo of mass is travelling on a constant longitude with speed . Find the magnitude and direction of the sideways component of apparent force experienced by the ski-doo. [The sideways component is locally along the surface of the Earth and perpendicular to the motion of the ski-doo.]
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Paper 3, Section I, D
2019 commentProve that two elements of are conjugate if and only if they have the same cycle type.
Describe a condition on the centraliser (in ) of a permutation that ensures the conjugacy class of in is the same as the conjugacy class of in . Justify your answer.
How many distinct conjugacy classes are there in ?
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Paper 3, Section , D
2019 commentWhat is the orthogonal group ? What is the special orthogonal group
Show that every element of has an eigenvector with eigenvalue
Is it true that every element of is either a rotation or a reflection? Justify your answer.
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Paper 3, Section II, D
2019 commentLet and be subgroups of a group satisfying the following two properties.
(i) All elements of can be written in the form for some and some .
(ii) .
Prove that and are normal subgroups of if and only if all elements of commute with all elements of .
State and prove Cauchy's Theorem.
Let and be distinct primes. Prove that an abelian group of order is isomorphic to . Is it true that all abelian groups of order are isomorphic to ?
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Paper 3, Section II, D
2019 commentState and prove Lagrange's Theorem.
Hence show that if is a finite group and then the order of divides the order of .
How many elements are there of order 3 in the following groups? Justify your answers.
(a) , where denotes the cyclic group of order .
(b) the dihedral group of order .
(c) the symmetric group of degree 7 .
(d) the alternating group of degree 7 .
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Paper 3, Section II, D
2019 commentState and prove the first isomorphism theorem. [You may assume that kernels of homomorphisms are normal subgroups and images are subgroups.]
Let be a group with subgroup and normal subgroup . Prove that is a subgroup of and is a normal subgroup of . Further, show that is a normal subgroup of .
Prove that is isomorphic to .
If and are both normal subgroups of must be a normal subgroup of ?
If and are subgroups of , but not normal subgroups, must be a subgroup of ?
Justify your answers.
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Paper 3, Section II, D
2019 commentLet be the group of Möbius transformations of and let be the group of all complex matrices of determinant 1 .
Show that the map given by
is a surjective homomorphism. Find its kernel.
Show that any not equal to the identity is conjugate to a Möbius map where either with or . [You may use results about matrices in as long as they are clearly stated.]
Show that any non-identity Möbius map has one or two fixed points. Also show that if is a Möbius map with just one fixed point then as for any . [You may assume that Möbius maps are continuous.]
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Paper 4, Section I, E
2019 commentFind all solutions to the simultaneous congruences
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Paper 4, Section , E
2019 commentShow that the series
converge. Determine in each case whether the limit is a rational number. Justify your answers.
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Paper 4, Section II, E
2019 comment(a) State and prove Fermat's theorem. Use it to compute .
(b) The Fibonacci numbers are defined by , and for all . Prove by induction that for all we have
(c) Let and let be an odd prime dividing . Which of the following statements are true, and which can be false? Justify your answers.
(i) If is odd then .
(ii) If is even then .
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Paper 4, Section II, E
2019 commentState the inclusion-exclusion principle.
Let be an integer. Let and
where is the largest number dividing all of . Let be the relation on where if .
(a) Show that
where the product is over all primes dividing .
(b) Show that if then there exist integers with .
(c) Show that if then there exists an integer with and . [Hint: Consider , where are as in part (b).] Deduce that is an equivalence relation.
(d) What is the size of the equivalence class containing Show that all equivalence classes have the same size, and deduce that the number of equivalence classes is
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Paper 4, Section II,
2019 comment(a) Let be a function. Show that the following statements are equivalent.
(i) is injective.
(ii) For every subset we have .
(iii) For every pair of subsets we have .
(b) Let be an injection. Show that for some subsets such that
[Here denotes the -fold composite of with itself.]
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Paper 4, Section II, E
2019 comment(a) What is a countable set? Let be sets with countable. Show that if is an injection then is countable. Deduce that and are countable. Show too that a countable union of countable sets is countable.
(b) Show that, in the plane, any collection of pairwise disjoint circles with rational radius is countable.
(c) A lollipop is any subset of the plane obtained by translating, rotating and scaling (by any factor ) the set
What happens if in part (b) we replace 'circles with rational radius' by 'lollipops'?
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Paper 2, Section I, 3F
2019 comment(a) Prove that as .
(b) State Stirling's approximation for !.
(c) A school party of boys and girls travel on a red bus and a green bus. Each bus can hold children. The children are distributed at random between the buses.
Let be the event that the boys all travel on the red bus and the girls all travel on the green bus. Show that
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Paper 2, Section I, F
2019 commentLet and be independent exponential random variables each with parameter 1 . Write down the joint density function of and .
Let and . Find the joint density function of and .
Are and independent? Briefly justify your answer.
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Paper 2, Section II, F
2019 comment(a) State the axioms that must be satisfied by a probability measure on a probability space .
Let and be events with . Define the conditional probability .
Let be pairwise disjoint events with for all and . Starting from the axioms, show that
and deduce Bayes' theorem.
(b) Two identical urns contain white balls and black balls. Urn I contains 45 white balls and 30 black balls. Urn II contains 12 white balls and 36 black balls. You do not know which urn is which.
(i) Suppose you select an urn and draw one ball at random from it. The ball is white. What is the probability that you selected Urn I?
(ii) Suppose instead you draw one ball at random from each urn. One of the balls is white and one is black. What is the probability that the white ball came from Urn I?
(c) Now suppose there are identical urns containing white balls and black balls, and again you do not know which urn is which. Each urn contains 1 white ball. The th urn contains black balls . You select an urn and draw one ball at random from it. The ball is white. Let be the probability that if you replace this ball and again draw a ball at random from the same urn then the ball drawn on the second occasion is also white. Show that as
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Paper 2, Section II, F
2019 commentLet and be positive integers with and let be a real number. A random walk on the integers starts at . At each step, the walk moves up 1 with probability and down 1 with probability . Find, with proof, the probability that the walk hits before it hits 0 .
Patricia owes a very large sum !) of money to a member of a violent criminal gang. She must return the money this evening to avoid terrible consequences but she only has !. She goes to a casino and plays a game with the probability of her winning being . If she bets on the game and wins then her is returned along with a further ; if she loses then her is lost.
The rules of the casino allow Patricia to play the game repeatedly until she runs out of money. She may choose the amount that she bets to be any integer a with , but it must be the same amount each time. What choice of would be best and why?
What choice of would be best, and why, if instead the probability of her winning the game is ?
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Paper 2, Section II, F
2019 commentRecall that a random variable in is bivariate normal or Gaussian if is normal for all . Let be bivariate normal.
(a) (i) Show that if is a real matrix then is bivariate normal.
(ii) Let and . Find the moment generating function of and deduce that the distribution of a bivariate normal random variable is uniquely determined by and .
(iii) Let and for . Let be the correlation of and . Write down in terms of some or all of and . If , why must and be independent?
For each , find . Hence show that for some normal random variable in that is independent of and some that should be specified.
(b) A certain species of East Anglian goblin has left arm of mean length with standard deviation , and right arm of mean length with standard deviation . The correlation of left- and right-arm-length of a goblin is . You may assume that the distribution of left- and right-arm-lengths can be modelled by a bivariate normal distribution. What is the probability that a randomly selected goblin has longer right arm than left arm?
[You may give your answer in terms of the distribution function of a random variable . That is, .J
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Paper 2, Section II, F
2019 commentLet be events in some probability space. Let be the number of that occur (so is a random variable). Show that
and
[Hint: Write where .]
A collection of lightbulbs are arranged in a circle. Each bulb is on independently with probability . Let be the number of bulbs such that both that bulb and the next bulb clockwise are on. Find and .
Let be the event that there is at least one pair of adjacent bulbs that are both on.
Use Markov's inequality to show that if then as .
Use Chebychev's inequality to show that if then as .
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Paper 3, Section I, B
2019 commentApply the divergence theorem to the vector field where is an arbitrary constant vector and is a scalar field, to show that
where is a volume bounded by the surface and is the outward pointing surface element.
Verify that this result holds when and is the spherical volume . [You may use the result that , where and are the usual angular coordinates in spherical polars and the components of are with respect to standard Cartesian axes.]
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Paper 3, Section I, B
2019 commentLet
Show that is an exact differential, clearly stating any criteria that you use.
Show that for any path between and
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Paper 3, Section II, B
2019 commentDefine the Jacobian, , of the one-to-one transformation
Give a careful explanation of the result
where
and the region maps under the transformation to the region .
Consider the region defined by
and
where and are positive constants.
Let be the intersection of with the plane . Write down the conditions for to be non-empty. Sketch the geometry of in , clearly specifying the curves that define its boundaries and points that correspond to minimum and maximum values of and of on the boundaries.
Use a suitable change of variables to evaluate the volume of the region , clearly explaining the steps in your calculation.
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Paper 3, Section II, B
2019 commentFor a given set of coordinate axes the components of a 2 nd rank tensor are given by .
(a) Show that if is an eigenvalue of the matrix with elements then it is also an eigenvalue of the matrix of the components of in any other coordinate frame.
Show that if is a symmetric tensor then the multiplicity of the eigenvalues of the matrix of components of is independent of coordinate frame.
A symmetric tensor in three dimensions has eigenvalues , with .
Show that the components of can be written in the form
where are the components of a unit vector.
(b) The tensor is defined by
where is the surface of the unit sphere, is the position vector of a point on , and is a constant.
Deduce, with brief reasoning, that the components of can be written in the form (1) with . [You may quote any results derived in part (a).]
Using suitable spherical polar coordinates evaluate and .
Explain how to deduce the values of and from and . [You do not need to write out the detailed formulae for these quantities.]
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Paper 3, Section II, B
2019 commentShow that for a vector field
Hence find an , with , such that . [Hint: Note that is not defined uniquely. Choose your expression for to be as simple as possible.
Now consider the cone . Let be the curved part of the surface of the cone and be the flat part of the surface of the cone .
Using the variables and as used in cylindrical polars to describe points on , give an expression for the surface element in terms of and .
Evaluate .
What does the divergence theorem predict about the two surface integrals and where in each case the vector is taken outwards from the cone?
What does Stokes theorem predict about the integrals and (defined as in the previous paragraph) and the line integral where is the circle and the integral is taken in the anticlockwise sense, looking from the positive direction?
Evaluate and , making your method clear and verify that each of these predictions holds.
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Paper 3, Section II, B
2019 comment(a) The function satisfies in the volume and on , the surface bounding .
Show that everywhere in .
The function satisfies in and is specified on . Show that for all functions such that on
Hence show that
(b) The function satisfies in the spherical region , with on . The function is spherically symmetric, i.e. .
Suppose that the equation and boundary conditions are satisfied by a spherically symmetric function . Show that
Hence find the function when is given by , with constant.
Explain how the results obtained in part (a) of the question imply that is the only solution of which satisfies the specified boundary condition on .
Use your solution and the results obtained in part (a) of the question to show that, for any function such that on and on ,
where is the region .
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Paper 1, Section I,
2019 comment(a) If
where , what is the value of ?
(b) Evaluate
(c) Find a complex number such that
(d) Interpret geometrically the curve defined by the set of points satisfying
in the complex -plane.
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Paper 1, Section I, A
2019 commentIf is an by matrix, define its determinant .
Find the following in terms of and a scalar , clearly showing your argument:
(i) , where is obtained from by multiplying one row by .
(ii) .
(iii) , where is obtained from by switching row and row .
(iv) , where is obtained from by adding times column to column .
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Paper 1, Section II, C
2019 comment(a) Use index notation to prove .
Hence simplify
(i) ,
(ii) .
(b) Give the general solution for and of the simultaneous equations
Show in particular that and must lie at opposite ends of a diameter of a sphere whose centre and radius should be found.
(c) If two pairs of opposite edges of a tetrahedron are perpendicular, show that the third pair are also perpendicular to each other. Show also that the sum of the lengths squared of two opposite edges is the same for each pair.
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Paper 1, Section II,
2019 commentLet be the standard basis vectors of . A second set of vectors are defined with respect to the standard basis by
The are the elements of the matrix . State the condition on under which the set forms a basis of .
Define the matrix that, for a given linear transformation , gives the relation between the components of any vector and those of the corresponding , with the components specified with respect to the standard basis.
Show that the relation between the matrix and the matrix of the same transformation with respect to the second basis is
Consider the matrix
Find a matrix such that is diagonal. Give the elements of and demonstrate explicitly that the relation between and holds.
Give the elements of for any positive integer .
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Paper 1, Section II, 7B
2019 comment(a) Let be an matrix. Define the characteristic polynomial of . [Choose a sign convention such that the coefficient of in the polynomial is equal to State and justify the relation between the characteristic polynomial and the eigenvalues of . Why does have at least one eigenvalue?
(b) Assume that has distinct eigenvalues. Show that . [Each term in corresponds to a term in
(c) For a general matrix and integer , show that , where Hint: You may find it helpful to note the factorization of .]
Prove that if has an eigenvalue then has an eigenvalue where .
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Paper 1, Section II, A
2019 commentThe exponential of a square matrix is defined as
where is the identity matrix. [You do not have to consider issues of convergence.]
(a) Calculate the elements of and , where
and is a real number.
(b) Show that and that
(c) Consider the matrices
Calculate:
(i) ,
(ii) .
(d) Defining
find the elements of the following matrices, where is a natural number:
(i)
(ii)
[Your answers to parts and should be in closed form, i.e. not given as series.]
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Paper 3, Section I,
2019 comment(a) Let . What does it mean for a function to be uniformly continuous?
(b) Which of the following functions are uniformly continuous? Briefly justify your answers.
(i) on .
(ii) on .
(iii) on .
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Paper 4, Section I, E
2019 commentLet . What does it mean to say that a sequence of real-valued functions on is uniformly convergent?
(i) If a sequence of real-valued functions on converges uniformly to , and each is continuous, must also be continuous?
(ii) Let . Does the sequence converge uniformly on ?
(iii) If a sequence of real-valued functions on converges uniformly to , and each is differentiable, must also be differentiable?
Give a proof or counterexample in each case.
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Paper 2, Section I, E
2019 commentConsider the map given by
where denotes the unique real cube root of .
(a) At what points is continuously differentiable? Calculate its derivative there.
(b) Show that has a local differentiable inverse near any with .
You should justify your answers, stating accurately any results that you require.
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Paper 1, Section II, E
2019 commentLet be an open subset. State what it means for a function to be differentiable at a point , and define its derivative .
State and prove the chain rule for the derivative of , where is a differentiable function.
Let be the vector space of real-valued matrices, and the open subset consisting of all invertible ones. Let be given by .
(a) Show that is differentiable at the identity matrix, and calculate its derivative.
(b) For , let be given by and . Show that on . Hence or otherwise, show that is differentiable at any point of , and calculate for .
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Paper 4, Section II, E
2019 comment(a) (i) Show that a compact metric space must be complete.
(ii) If a metric space is complete and bounded, must it be compact? Give a proof or counterexample.
(b) A metric space is said to be totally bounded if for all , there exists and such that
(i) Show that a compact metric space is totally bounded.
(ii) Show that a complete, totally bounded metric space is compact.
[Hint: If is Cauchy, then there is a subsequence such that
(iii) Consider the space of continuous functions , with the metric
Is this space compact? Justify your answer.
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Paper 3, Section II, E
2019 comment(a) Carefully state the Picard-Lindelöf theorem on solutions to ordinary differential equations.
(b) Let be the set of continuous functions from a closed interval to , and let be a norm on .
(i) Let . Show that for any the norm
is Lipschitz equivalent to the usual sup norm on .
(ii) Assume that is continuous and Lipschitz in the second variable, i.e. there exists such that
for all and all . Define by
for .
Show that there is a choice of such that is a contraction on . Deduce that for any , the differential equation
has a unique solution on with .
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Paper 2, Section II, 12E
2019 comment(a) (i) Define what it means for two norms on a vector space to be Lipschitz equivalent.
(ii) Show that any two norms on a finite-dimensional vector space are Lipschitz equivalent.
(iii) Show that if two norms on a vector space are Lipschitz equivalent then the following holds: for any sequence in is Cauchy with respect to if and only if it is Cauchy with respect to .
(b) Let be the vector space of real sequences such that . Let
and for , let
You may assume that and are well-defined norms on .
(i) Show that is not Lipschitz equivalent to for any .
(ii) Are there any with such that and are Lipschitz equivalent? Justify your answer.
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Paper 4, Section I,
2019 commentState the Cauchy Integral Formula for a disc. If is a holomorphic function such that for all , show using the Cauchy Integral Formula that is constant.
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Paper 3, Section II, F
2019 commentDefine the winding number of a closed path around a point which does not lie on the image of . [You do not need to justify its existence.]
If is a meromorphic function, define the order of a zero of and of a pole of . State the Argument Principle, and explain how it can be deduced from the Residue Theorem.
How many roots of the polynomial
lie in the right-hand half plane?
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Paper 1, Section I, F
2019 commentWhat is the Laurent series for a function defined in an annulus ? Find the Laurent series for on the annuli
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Paper 1, Section II, F
2019 commentState and prove Jordan's lemma.
What is the residue of a function at an isolated singularity ? If with a positive integer, analytic, and , derive a formula for the residue of at in terms of derivatives of .
Evaluate
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Paper 2, Section II, D
2019 commentLet and be smooth curves in the complex plane, intersecting at some point . Show that if the map is complex differentiable, then it preserves the angle between and at , provided . Give an example that illustrates why the condition is important.
Show that is a one-to-one conformal map on each of the two regions and , and find the image of each region.
Hence construct a one-to-one conformal map from the unit disc to the complex plane with the intervals and removed.
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Paper 3, Section I, D
2019 commentBy considering the transformation , find a solution to Laplace's equation inside the unit disc , subject to the boundary conditions
where is constant. Give your answer in terms of .
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Paper 4, Section II, D
2019 comment(a) Using the Bromwich contour integral, find the inverse Laplace transform of .
The temperature of mercury in a spherical thermometer bulb obeys the radial heat equation
with unit diffusion constant. At the mercury is at a uniform temperature equal to that of the surrounding air. For the surrounding air temperature lowers such that at the edge of the thermometer bulb
where is a constant.
(b) Find an explicit expression for .
(c) Show that the temperature of the mercury at the centre of the thermometer bulb at late times is
[You may assume that the late time behaviour of is determined by the singular part of at
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Paper 2, Section I, A
2019 commentWrite down the solution for the scalar potential that satisfies
with as . You may assume that the charge distribution vanishes for , for some constant . In an expansion of for , show that the terms of order and can be expressed in terms of the total charge and the electric dipole moment , which you should define.
Write down the analogous solution for the vector potential that satisfies
with as . You may assume that the current vanishes for and that it obeys everywhere. In an expansion of for , show that the term of order vanishes.
Hint:
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Paper 4, Section I, A
2019 commentWrite down Maxwell's Equations for electric and magnetic fields and in the absence of charges and currents. Show that there are solutions of the form
if and satisfy a constraint and if and are then chosen appropriately.
Find the solution with , where is real, and . Compute the Poynting vector and state its physical significance.
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Paper 1, Section II, A
2019 commentLet be the electric field and the scalar potential due to a static charge density , with all quantities vanishing as becomes large. The electrostatic energy of the configuration is given by
with the integrals taken over all space. Verify that these integral expressions agree.
Suppose that a total charge is distributed uniformly in the region and that otherwise. Use the integral form of Gauss's Law to determine at all points in space and, without further calculation, sketch graphs to indicate how and depend on position.
Consider the limit with fixed. Comment on the continuity of and . Verify directly from each of the integrals in that in this limit.
Now consider a small change in the total charge . Show that the first-order change in the energy is and interpret this result.
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Paper 3, Section II, A
2019 commentThe electric and magnetic fields in an inertial frame are related to the fields in a frame by a Lorentz transformation. Given that moves in the -direction with speed relative to , and that
write down equations relating the remaining field components and define . Use your answers to show directly that .
Give an expression for an additional, independent, Lorentz-invariant function of the fields, and check that it is invariant for the special case when and are the only non-zero components in the frame .
Now suppose in addition that with a non-zero constant. Show that the angle between the electric and magnetic fields in is given by
where . By considering the behaviour of as approaches its limiting values, show that the relative velocity of the frames can be chosen so that the angle takes any value in one of the ranges or , depending on the sign of .
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Paper 2, Section II, A
2019 commentConsider a conductor in the shape of a closed curve moving in the presence of a magnetic field B. State Faraday's Law of Induction, defining any quantities that you introduce.
Suppose is a square horizontal loop that is allowed to move only vertically. The location of the loop is specified by a coordinate , measured vertically upwards, and the edges of the loop are defined by and . If the magnetic field is
where is a constant, find the induced current , given that the total resistance of the loop is .
Calculate the resulting electromagnetic force on the edge of the loop , and show that this force acts at an angle to the vertical. Find the total electromagnetic force on the loop and comment on its direction.
Now suppose that the loop has mass and that gravity is the only other force acting on it. Show that it is possible for the loop to fall with a constant downward velocity .
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Paper 1, Section I, C
2019 commentA viscous fluid flows steadily down a plane that is inclined at an angle to the horizontal. The fluid layer is of uniform thickness and has a free upper surface. Determine the velocity profile in the direction perpendicular to the plane and also the volume flux (per unit width), in terms of the gravitational acceleration , the angle , the kinematic viscosity and the thickness of the fluid layer.
Show that the volume flux is reduced if the free upper surface is replaced by a stationary plane boundary, and give a physical explanation for this.
-
Paper 2, Section I, C
2019 commentConsider the steady flow
where are Cartesian coordinates. Show that and determine the streamfunction. Calculate the vorticity and verify that the vorticity equation is satisfied in the absence of viscosity. Sketch the streamlines in the region .
-
Paper 1, Section II, C
2019 commentExplain why the irrotational flow of an incompressible fluid can be expressed in terms of a velocity potential that satisfies Laplace's equation.
The axis of a stationary cylinder of radius coincides with the -axis of a Cartesian coordinate system with unit vectors . A fluid of density flows steadily past the cylinder such that the velocity field is independent of and has no component in the -direction. The flow is irrotational but there is a constant non-zero circulation
around every closed curve that encloses the cylinder once in a positive sense. Far from the cylinder, the velocity field tends towards the uniform flow , where is a constant.
State the boundary conditions on the velocity potential, in terms of polar coordinates in the -plane. Explain why the velocity potential is not required to be a single-valued function of position. Hence obtain the appropriate solution , in terms of and .
Neglecting gravity, show that the net force on the cylinder, per unit length in the -direction, is
Determine the number and location of stagnation points in the flow as a function of the dimensionless parameter
-
Paper 4, Section II, C
2019 commentThe linear shallow-water equations governing the motion of a fluid layer in the neighbourhood of a point on the Earth's surface in the northern hemisphere are
where and are the horizontal velocity components and is the perturbation of the height of the free surface.
(a) Explain the meaning of the three positive constants and appearing in the equations above and outline the assumptions made in deriving these equations.
(b) Show that , the -component of vorticity, satisfies
and deduce that the potential vorticity
satisfies
(c) Consider a steady geostrophic flow that is uniform in the latitudinal direction. Show that
Given that the potential vorticity has the piecewise constant profile
where and are constants, and that as , solve for and in terms of the Rossby radius . Sketch the functions and in the case .
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Paper 3, Section II, C
2019 commentA cubic box of side , enclosing the region , contains equal volumes of two incompressible fluids that remain distinct. The system is initially at rest, with the fluid of density occupying the region and the fluid of density occupying the region , and with gravity . The interface between the fluids is then slightly perturbed. Derive the linearized equations and boundary conditions governing small disturbances to the initial state.
In the case , show that the angular frequencies of the normal modes are given by
and express the allowable values of the wavenumber in terms of . Identify the lowestfrequency non-trivial mode . Comment on the limit . What physical behaviour is expected in the case ?
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Paper 1, Section I, E
2019 commentDescribe the Poincaré disc model for the hyperbolic plane by giving the appropriate Riemannian metric.
Calculate the distance between two points . You should carefully state any results about isometries of that you use.
-
Paper 3, Section I, E
2019 commentState a formula for the area of a spherical triangle with angles .
Let . What is the area of a convex spherical -gon with interior angles ? Justify your answer.
Find the range of possible values for the interior angle of a regular convex spherical
-
Paper 3, Section II, E
2019 commentDefine a geodesic triangulation of an abstract closed smooth surface. Define the Euler number of a triangulation, and state the Gauss-Bonnet theorem for closed smooth surfaces. Given a vertex in a triangulation, its valency is defined to be the number of edges incident at that vertex.
(a) Given a triangulation of the torus, show that the average valency of a vertex of the triangulation is 6 .
(b) Consider a triangulation of the sphere.
(i) Show that the average valency of a vertex is strictly less than 6 .
(ii) A triangulation can be subdivided by replacing one triangle with three sub-triangles, each one with vertices two of the original ones, and a fixed interior point of .

Using this, or otherwise, show that there exist triangulations of the sphere with average vertex valency arbitrarily close to 6 .
(c) Suppose is a closed abstract smooth surface of everywhere negative curvature. Show that the average vertex valency of a triangulation of is bounded above and below.
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Paper 2, Section II, E
2019 commentDefine a smooth embedded surface in . Sketch the surface given by
and find a smooth parametrisation for it. Use this to calculate the Gaussian curvature of at every point.
Hence or otherwise, determine which points of the embedded surface
have Gaussian curvature zero. [Hint: consider a transformation of .]
[You should carefully state any result that you use.]
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Paper 4, Section II, E
2019 commentLet be the upper-half plane with hyperbolic metric . Define the group , and show that it acts by isometries on . [If you use a generation statement you must carefully state it.]
(a) Prove that acts transitively on the collection of pairs , where is a hyperbolic line in and .
(b) Let be the imaginary half-axis. Find the isometries of which fix pointwise. Hence or otherwise find all isometries of .
(c) Describe without proof the collection of all hyperbolic lines which meet with (signed) angle . Explain why there exists a hyperbolic triangle with angles and whenever .
(d) Is this triangle unique up to isometry? Justify your answer. [You may use without proof the fact that Möbius maps preserve angles.]
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Paper 3, Section I,
2019 commentProve that the ideal in is not principal.
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Paper 4, Section I, G
2019 commentLet be a group and a subgroup.
(a) Define the normaliser .
(b) Suppose that and is a Sylow -subgroup of . Using Sylow's second theorem, prove that .
-
Paper 2, Section I, G
2019 commentLet be an integral domain. A module over is torsion-free if, for any and only if or .
Let be a module over . Prove that there is a quotient
with torsion-free and with the following property: whenever is a torsion-free module and is a homomorphism of modules, there is a homomorphism such that .
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Paper 1, Section II, G
2019 comment(a) Let be a group of order , for a prime. Prove that is not simple.
(b) State Sylow's theorems.
(c) Let be a group of order , where are distinct odd primes. Prove that is not simple.
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Paper 4, Section II, G
2019 comment(a) Define the Smith Normal Form of a matrix. When is it guaranteed to exist?
(b) Deduce the classification of finitely generated abelian groups.
(c) How many conjugacy classes of matrices are there in with minimal polynomial
-
Paper 3, Section II, G
2019 commentLet .
(a) Prove that is a Euclidean domain.
(b) Deduce that is a unique factorisation domain, stating carefully any results from the course that you use.
(c) By working in , show that whenever satisfy
then is not congruent to 2 modulo 3 .
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Paper 2, Section II, G
2019 comment(a) Let be a field and let be an irreducible polynomial of degree over . Prove that there exists a field containing as a subfield such that
where and . State carefully any results that you use.
(b) Let be a field and let be a monic polynomial of degree over , which is not necessarily irreducible. Prove that there exists a field containing as a subfield such that
where .
(c) Let for a prime, and let for an integer. For as in part (b), let be the set of roots of in . Prove that is a field.
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Paper 4, Section I, F
2019 commentWhat is an eigenvalue of a matrix ? What is the eigenspace corresponding to an eigenvalue of ?
Consider the matrix
for a non-zero vector. Show that has rank 1 . Find the eigenvalues of and describe the corresponding eigenspaces. Is diagonalisable?
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Paper 2, Section I, F
2019 commentIf and are finite-dimensional subspaces of a vector space , prove that
Let
Show that is 3 -dimensional and find a linear map such that
-
Paper 1, Section I, F
2019 commentDefine a basis of a vector space .
If has a finite basis , show using only the definition that any other basis has the same cardinality as .
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Paper 1, Section II, F
2019 commentWhat is the adjugate adj of an matrix ? How is it related to
(a) Define matrices by
and scalars by
Find a recursion for the matrices in terms of and the 's.
(b) By considering the partial derivatives of the multivariable polynomial
show that
(c) Hence show that the 's may be expressed in terms of .
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Paper 4, Section II, F
2019 commentIf is a finite-dimensional real vector space with inner product , prove that the linear map given by is an isomorphism. [You do not need to show that it is linear.]
If and are inner product spaces and is a linear map, what is meant by the adjoint of ? If is an orthonormal basis for is an orthonormal basis for , and is the matrix representing in these bases, derive a formula for the matrix representing in these bases.
Prove that .
If then the linear equation has no solution, but we may instead search for a minimising , known as a least-squares solution. Show that is such a least-squares solution if and only if it satisfies . Hence find a least-squares solution to the linear equation
-
Paper 3, Section II, F
2019 commentIf is a quadratic form on a finite-dimensional real vector space , what is the associated symmetric bilinear form ? Prove that there is a basis for with respect to which the matrix for is diagonal. What is the signature of ?
If is a subspace such that for all and all , show that defines a quadratic form on the quotient vector space . Show that the signature of is the same as that of .
If are vectors such that and , show that there is a direct sum decomposition such that the signature of is the same as that of .
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Paper 2, Section II, F
2019 commentLet and be matrices over .
(a) Assuming that is invertible, show that and have the same characteristic polynomial.
(b) By considering the matrices , show that and have the same characteristic polynomial even when is singular.
(c) Give an example to show that the minimal polynomials and of and may be different.
(d) Show that and differ at most by a factor of . Stating carefully any results which you use, deduce that if is diagonalisable then so is .
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Paper 4, Section I, H
2019 commentFor a Markov chain on a state space with , we let for be the probability that when .
(a) Let be a Markov chain. Prove that if is recurrent at a state , then . [You may use without proof that the number of returns of a Markov chain to a state when starting from has the geometric distribution.]
(b) Let and be independent simple symmetric random walks on starting from the origin 0 . Let . Prove that and deduce that . [You may use without proof that for all and , and that is recurrent at 0.]
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Paper 3, Section I, H
2019 commentSuppose that is a Markov chain with state space .
(a) Give the definition of a communicating class.
(b) Give the definition of the period of a state .
(c) Show that if two states communicate then they have the same period.
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Paper 2, Section II, H
2019 commentFix and let be the graph consisting of a copy of joining vertices and , a copy of joining vertices and , and a copy of joining vertices and . Let be the vertex adjacent to on the segment from to . Shown below is an illustration of in the case . The vertices are solid squares and edges are indicated by straight lines.

Let be a simple random walk on . In other words, in each time step, moves to one of its neighbours with equal probability. Assume that .
(a) Compute the expected amount of time for to hit .
(b) Compute the expected amount of time for to hit . [Hint: first show that the expected amount of time for to go from to satisfies where is the expected return time of to when starting from .]
(c) Compute the expected amount of time for to hit . [Hint: for each , let be the vertex which is places to the right of on the segment from to . Derive an equation for the expected amount of time for to go from to .]
Justify all of your answers.
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Paper 1, Section II, H
2019 commentLet be a transition matrix for a Markov chain on a state space with elements with . Assume that the Markov chain is aperiodic and irreducible and let be its unique invariant distribution. Assume that .
(a) Let . Show that .
(b) Let . Compute in terms of an explicit function of .
(c) Suppose that a cop and a robber start from a common state chosen from . The robber then takes one step according to and stops. The cop then moves according to independently of the robber until the cop catches the robber (i.e., the cop visits the state occupied by the robber). Compute the expected amount of time for the cop to catch the robber.
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Paper 2, Section I, B
2019 commentLet be spherical polar coordinates, and let denote the th Legendre polynomial. Write down the most general solution for of Laplace's equation that takes the form .
Solve Laplace's equation in the spherical shell subject to the boundary conditions
[The first three Legendre polynomials are
-
Paper 4, Section I, D
2019 commentLet
By considering the integral , where is a smooth, bounded function that vanishes sufficiently rapidly as , identify in terms of a generalized function.
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Paper 3, Section I, D
2019 commentDefine the discrete Fourier transform of a sequence of complex numbers.
Compute the discrete Fourier transform of the sequence
-
Paper 1, Section II, B
2019 commentThe Bessel functions can be defined by the expansion
By using Cartesian coordinates , or otherwise, show that
Deduce that satisfies Bessel's equation
By expanding the left-hand side of up to cubic order in , derive the series expansions of and up to this order.
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Paper 3, Section II, D
2019 commentBy differentiating the expression , where is a constant and is the Heaviside step function, show that
where is the Dirac -function.
Hence, by taking a Fourier transform with respect to the spatial variables only, derive the retarded Green's function for the wave operator in three spatial dimensions.
[You may use that
without proof.]
Thus show that the solution to the homogeneous wave equation , subject to the initial conditions and , may be expressed as
where is the average value of on a sphere of radius centred on . Interpret this result.
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Paper 2, Section II, D
2019 commentFor , the degree polynomial satisfies the differential equation
where is a real, positive parameter. Show that, when ,
for a weight function and values that you should determine.
Suppose that the roots of that lie inside the domain are , with . By considering the integral
show that in fact all roots of lie in .
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Paper 4, Section II, B
2019 comment(a) Show that the operator
where and are real functions, is self-adjoint (for suitable boundary conditions which you need not state) if and only if
(b) Consider the eigenvalue problem
on the interval with boundary conditions
Assuming that is everywhere negative, show that all eigenvalues are positive.
(c) Assume now that and that the eigenvalue problem (*) is on the interval with . Show that is an eigenvalue provided that
and show graphically that this condition has just one solution in the range .
[You may assume that all eigenfunctions are either symmetric or antisymmetric about
-
Paper 3, Section I,
2019 commentLet be a metric space.
(a) What does it mean for to be compact? What does it mean for to be sequentially compact?
(b) Prove that if is compact then is sequentially compact.
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Paper 2, Section I, G
2019 comment(a) Let be a continuous surjection of topological spaces. Prove that, if is connected, then is also connected.
(b) Let be a continuous map. Deduce from part (a) that, for every between and , there is such that . [You may not assume the Intermediate Value Theorem, but you may use the fact that suprema exist in .]
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Paper 1, Section II, G
2019 commentConsider the set of sequences of integers
Define
for two sequences . Let
where, as usual, we adopt the convention that .
(a) Prove that defines a metric on .
(b) What does it mean for a metric space to be complete? Prove that is complete.
(c) Is path connected? Justify your answer.
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Paper 4, Section II, G
2019 comment(a) Define the subspace, quotient and product topologies.
(b) Let be a compact topological space and a Hausdorff topological space. Prove that a continuous bijection is a homeomorphism.
(c) Let , equipped with the product topology. Let be the smallest equivalence relation on such that and , for all . Let
equipped with the subspace topology from . Prove that and are homeomorphic.
[You may assume without proof that is compact.]
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Paper 1, Section I, C
2019 commentLet be the smallest interval that contains the distinct real numbers , and let be a continuous function on that interval.
Define the divided difference of degree .
Prove that the polynomial of degree that interpolates the function at the points is equal to the Newton polynomial
Prove the recursive formula
for
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Paper 4, Section I, C
2019 commentCalculate the factorization of the matrix
Use this to evaluate and to solve the equation
with
-
Paper 1, Section II, C
2019 comment(a) An -step method for solving the ordinary differential equation
is given by
where and are constant coefficients, with , and is the time-step. Prove that the method is of order if and only if
as , where
(b) Show that the Adams-Moulton method
is of third order and convergent.
[You may assume the Dahlquist equivalence theorem if you state it clearly.]
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Paper 3, Section II, C
2019 comment(a) Let be a positive weight function on the interval . Show that
defines an inner product on .
(b) Consider the sequence of polynomials defined by the three-term recurrence relation
where
and the coefficients (for and (for are given by
Prove that this defines a sequence of monic orthogonal polynomials on .
(c) The Hermite polynomials are orthogonal on the interval with weight function . Given that
deduce that the Hermite polynomials satisfy a relation of the form with and . Show that .
(d) State, without proof, how the properties of the Hermite polynomial , for some positive integer , can be used to estimate the integral
where is a given function, by the method of Gaussian quadrature. For which polynomials is the quadrature formula exact?
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Paper 2, Section II, C
2019 commentDefine the linear least squares problem for the equation
where is a given matrix with is a given vector and is an unknown vector.
Explain how the linear least squares problem can be solved by obtaining a factorization of the matrix , where is an orthogonal matrix and is an uppertriangular matrix in standard form.
Use the Gram-Schmidt method to obtain a factorization of the matrix
and use it to solve the linear least squares problem in the case
-
Paper 1, Section I, H
2019 commentSuppose that is an infinitely differentiable function on . Assume that there exist constants so that and for all . Fix and for each set
Let be the unique value of where attains its minimum. Prove that
[Hint: Express in terms of the Taylor series for at using the Lagrange form of the remainder: where is between and
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Paper 2, Section I, H
2019 commentState the Lagrange sufficiency theorem.
Find the maximum of over subject to the constraint
using Lagrange multipliers. Carefully justify why your solution is in fact the maximum.
Find the maximum of over subject to the constraint
-
Paper 4, Section II, H
2019 comment(a) State and prove the max-flow min-cut theorem.
(b) (i) Apply the Ford-Fulkerson algorithm to find the maximum flow of the network illustrated below, where is the source and is the sink.

(ii) Verify the optimality of your solution using the max-flow min-cut theorem.
(iii) Is there a unique flow which attains the maximum? Explain your answer.
(c) Prove that the Ford-Fulkerson algorithm always terminates when the network is finite, the capacities are integers, and the algorithm is initialised where the initial flow is 0 across all edges. Prove also in this case that the flow across each edge is an integer.
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Paper 3, Section II, H
2019 comment(a) Suppose that and , with . What does it mean for to be a basic feasible solution of the equation
Assume that the rows of are linearly independent, every set of columns is linearly independent, and every basic solution has exactly non-zero entries. Prove that the extreme points of are the basic feasible solutions of . [Here, means that each of the coordinates of are at least 0 .]
(b) Use the simplex method to solve the linear program
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Paper 4, Section I, B
2019 comment(a) Define the probability density and probability current for the wavefunction of a particle of mass . Show that
and deduce that for a normalizable, stationary state wavefunction. Give an example of a non-normalizable, stationary state wavefunction for which is non-zero, and calculate the value of .
(b) A particle has the instantaneous, normalized wavefunction
where is positive and is real. Calculate the expectation value of the momentum for this wavefunction.
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Paper 3, Section , B
2019 commentConsider a quantum mechanical particle moving in two dimensions with Cartesian coordinates . Show that, for wavefunctions with suitable decay as , the operators
are Hermitian, and similarly
are Hermitian.
Show that if and are Hermitian operators, then
is Hermitian. Deduce that
are Hermitian. Show that
-
Paper 1, Section II, B
2019 commentStarting from the time-dependent Schrödinger equation, show that a stationary state of a particle of mass in a harmonic oscillator potential in one dimension with frequency satisfies
Find a rescaling of variables that leads to the simplified equation
Setting , find the equation satisfied by .
Assume now that is a polynomial
Determine the value of and deduce the corresponding energy level of the harmonic oscillator. Show that if is even then the stationary state has even parity.
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Paper 3, Section II, B
2019 commentConsider a particle of unit mass in a one-dimensional square well potential
with infinite outside. Find all the stationary states and their energies , and write down the general normalized solution of the time-dependent Schrödinger equation in terms of these.
The particle is initially constrained by a barrier to be in the ground state in the narrower square well potential
with infinite outside. The barrier is removed at time , and the wavefunction is instantaneously unchanged. Show that the particle is now in a superposition of stationary states of the original potential well, and calculate the probability that an energy measurement will yield the result .
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Paper 2, Section II, B
2019 commentLet be Cartesian coordinates in . The angular momentum operators satisfy the commutation relation
and its cyclic permutations. Define the total angular momentum operator and show that . Write down the explicit form of .
Show that a function of the form , where , is an eigenfunction of and find the eigenvalue. State the analogous result for .
There is an energy level for a particle in a certain spherically symmetric potential well that is 6-fold degenerate. A basis for the (unnormalized) energy eigenstates is of the form
Find a new basis that consists of simultaneous eigenstates of and and identify their eigenvalues.
[You may quote the range of eigenvalues associated with a particular eigenvalue of .]
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Paper 1, Section I, H
2019 commentSuppose that are i.i.d. random variables.
(a) Compute the MLEs for the unknown parameters .
(b) Give the definition of an unbiased estimator. Determine whether are unbiased estimators for .
-
Paper 2, Section I, H
2019 commentSuppose that are i.i.d. coin tosses with probability of obtaining a head.
(a) Compute the posterior distribution of given the observations in the case of a uniform prior on .
(b) Give the definition of the quadratic error loss function.
(c) Determine the value of which minimizes the quadratic error loss function. Justify your answer. Calculate .
[You may use that the , distribution has density function on given by
where is a normalizing constant. You may also use without proof that the mean of a random variable is
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Paper 4, Section II, 19H
2019 commentConsider the linear model
where are known and are i.i.d. . We assume that the parameters and are unknown.
(a) Find the MLE of . Explain why is the same as the least squares estimator of .
(b) State and prove the Gauss-Markov theorem for this model.
(c) For each value of with , determine the unbiased linear estimator of which minimizes
-
Paper 1, Section II, H
2019 commentState and prove the Neyman-Pearson lemma.
Suppose that are i.i.d. random variables where is an unknown parameter. We wish to test the hypothesis against the hypothesis where .
(a) Find the critical region of the likelihood ratio test of size in terms of the sample mean .
(b) Define the power function of a hypothesis test and identify the power function in the setting described above in terms of the probability distribution function. [You may use without proof that is distributed as a random variable.]
(c) Define what it means for a hypothesis test to be uniformly most powerful. Determine whether the likelihood ratio test considered above is uniformly most powerful for testing against .
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Paper 3, Section II, H
2019 commentSuppose that are i.i.d. . Let
(a) Compute the distributions of and and show that and are independent.
(b) Write down the distribution of .
(c) For , find a confidence interval in each of the following situations: (i) for when is known; (ii) for when is not known; (iii) for when is not known.
(d) Suppose that are i.i.d. . Explain how you would use the test to test the hypothesis against the hypothesis . Does the test depend on whether are known?
-
Paper 1, Section I, A
2019 commentA function is defined on the surface . Find the location of every stationary point of this function.
-
Paper 3, Section I, A
2019 commentThe function with domain is defined by , where . Verify that is convex, using an appropriate sufficient condition.
Determine the Legendre transform of , specifying clearly its domain of definition, and find .
Show that
where and and are positive real numbers such that .
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Paper 2, Section II, A
2019 commentWrite down the Euler-Lagrange (EL) equations for a functional
where and each take specified values at and . Show that the EL equations imply that
is independent of provided satisfies a certain condition, to be specified. State conditions under which there exist additional first integrals of the equations.
Consider
where is a positive constant. Show that solutions of the EL equations satisfy
for some constant . Assuming that , find and hence determine the most general solution for as a function of subject to the conditions and as . Show that, for any such solution, as .
[Hint:
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Paper 4, Section II, A
2019 commentConsider the functional
where is subject to boundary conditions as with . [You may assume the integral converges.]
(a) Find expressions for the first-order and second-order variations and resulting from a variation that respects the boundary conditions.
(b) If , show that if and only if for all . Explain briefly how this is consistent with your results for and in part (a).
(c) Now suppose that with . By considering an integral of , show that
with equality if and only if satisfies a first-order differential equation. Deduce that global minima of with the specified boundary conditions occur precisely for
where is a constant. How is the first-order differential equation that appears in this case related to your general result for in part (a)?
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Paper 4, Section II, F
2019 comment(a) Let be a smooth projective plane curve, defined by a homogeneous polynomial of degree over the complex numbers .
(i) Define the divisor , where is a hyperplane in not contained in , and prove that it has degree .
(ii) Give (without proof) an expression for the degree of in terms of .
(iii) Show that does not have genus 2 .
(b) Let be a smooth projective curve of genus over the complex numbers . For let
there is no with , and for all
(i) Define , for a divisor .
(ii) Show that for all ,
(iii) Show that has exactly elements. [Hint: What happens for large ?]
(iv) Now suppose that has genus 2 . Show that or .
[In this question denotes the set of positive integers.]
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Paper 3, Section II, F
2019 commentLet be the curve defined by the equation over the complex numbers , and let be its closure.
(a) Show is smooth.
(b) Determine the ramification points of the defined by
Using this, determine the Euler characteristic and genus of , stating clearly any theorems that you are using.
(c) Let . Compute for all , and determine a basis for
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Paper 2, Section II, F
2019 comment(a) Let be a commutative algebra over a field , and a -linear homomorphism. Define , the derivations of centered in , and define the tangent space in terms of this.
Show directly from your definition that if is not a zero divisor and , then the natural map is an isomorphism.
(b) Suppose is an algebraically closed field and for . Let
Find a surjective map . Justify your answer.
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Paper 1, Section II, F
2019 comment(a) Let be an algebraically closed field of characteristic 0 . Consider the algebraic variety defined over by the polynomials
Determine
(i) the irreducible components of ,
(ii) the tangent space at each point of ,
(iii) for each irreducible component, the smooth points of that component, and
(iv) the dimensions of the irreducible components.
(b) Let be a finite extension of fields, and . Identify with over and show that
is the complement in of the vanishing set of some polynomial. [You need not show that is non-empty. You may assume that if and only if form a basis of over .]
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Paper 3, Section II, F
2019 commentLet be a simplicial complex, and a subcomplex. As usual, denotes the group of -chains of , and denotes the group of -chains of .
(a) Let
for each integer . Prove that the boundary map of descends to give the structure of a chain complex.
(b) The homology groups of relative to , denoted by , are defined to be the homology groups of the chain complex . Prove that there is a long exact sequence that relates the homology groups of relative to to the homology groups of and the homology groups of .
(c) Let be the closed -dimensional disc, and be the -dimensional sphere. Exhibit simplicial complexes and subcomplexes such that in such a way that is identified with .
(d) Compute the relative homology groups , for all integers and where and are as in (c).
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Paper 4, Section II, F
2019 commentState the Lefschetz fixed point theorem.
Let be an integer, and a choice of base point. Define a space
where is discrete and is the smallest equivalence relation such that for all . Let be a homeomorphism without fixed points. Use the Lefschetz fixed point theorem to prove the following facts.
(i) If then is divisible by 3 .
(ii) If then is even.
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Paper 2, Section II, F
2019 commentLet and . Let be the natural inclusion maps. Consider the space ; that is,
where is the smallest equivalence relation such that for all .
(a) Prove that is homeomorphic to the 3 -sphere .
[Hint: It may help to think of as contained in .]
(b) Identify as a quotient of the square in the usual way. Let be the circle in given by the equation is illustrated in the figure below.

Compute a presentation for , where is the complement of in , and deduce that is non-abelian.
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Paper 1, Section II, F
2019 commentIn this question, and are path-connected, locally simply connected spaces.
(a) Let be a continuous map, and a path-connected covering space of . State and prove a uniqueness statement for lifts of to .
(b) Let be a covering map. A covering transformation of is a homeomorphism such that . For each integer , give an example of a space and an -sheeted covering map such that the only covering transformation of is the identity map. Justify your answer. [Hint: Take to be a wedge of two circles.]
(c) Is there a space and a 2-sheeted covering map for which the only covering transformation of is the identity? Justify your answer briefly.
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Paper 3, Section II, H
2019 comment(a) Prove that in a finite-dimensional normed vector space the weak and strong topologies coincide.
(b) Prove that in a normed vector space , a weakly convergent sequence is bounded. [Any form of the Banach-Steinhaus theorem may be used, as long as you state it clearly.]
(c) Let be the space of real-valued absolutely summable sequences. Suppose is a weakly convergent sequence in which does not converge strongly. Show there is a constant and a sequence in which satisfies and for all .
With as above, show there is some and a subsequence of with for all . Deduce that every weakly convergent sequence in is strongly convergent.
[Hint: Define so that for , where the sequence of integers should be defined inductively along with
(d) Is the conclusion of part (c) still true if we replace by
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Paper 1, Section II, H
2019 comment(a) Consider the topology on the natural numbers induced by the standard topology on . Prove it is the discrete topology; i.e. is the power set of .
(b) Describe the corresponding Borel sets on and prove that any function or is measurable.
(c) Using Lebesgue integration theory, define for a function and then for . State any condition needed for the sum of the latter series to be defined. What is a simple function in this setting, and which simple functions have finite sum?
(d) State and prove the Beppo Levi theorem (also known as the monotone convergence theorem).
(e) Consider such that for any , the function is non-decreasing. Prove that
Show that this need not be the case if we drop the hypothesis that is nondecreasing, even if all the relevant limits exist.
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Paper 4, Section II, B
2019 comment(a) A classical beam of particles scatters off a spherically symmetric potential . Draw a diagram to illustrate the differential cross-section , and use this to derive an expression for in terms of the impact parameter and the scattering angle .
A quantum beam of particles of mass and momentum is incident along the -axis and scatters off a spherically symmetric potential . Write down the asymptotic form of the wavefunction in terms of the scattering amplitude . By considering the probability current , derive an expression for the differential cross-section in terms of .
(b) The solution of the radial Schrödinger equation for a particle of mass and wave number moving in a spherically symmetric potential has the asymptotic form
valid for , where and are constants and denotes the th Legendre polynomial. Define the S-matrix element and the corresponding phase shift for the partial wave of angular momentum , in terms of and . Define also the scattering length for the potential .
Outside some core region, , the Schrödinger equation for some such potential is solved by the s-wave (i.e. ) wavefunction with,
where is a constant. Extract the S-matrix element , the phase shift and the scattering length . Deduce that the potential has a bound state of zero angular momentum and compute its energy. Give the form of the (un-normalised) bound state wavefunction in the region .
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Paper 3, Section II, B
2019 commentA Hamiltonian is invariant under the discrete translational symmetry of a Bravais lattice . This means that there exists a unitary translation operator such that for all . State and prove Bloch's theorem for .
Consider the two-dimensional Bravais lattice defined by the basis vectors
Find basis vectors and for the reciprocal lattice. Sketch the Brillouin zone. Explain why the Brillouin zone has only two physically distinct corners. Show that the positions of these corners may be taken to be and .
The dynamics of a single electron moving on the lattice is described by a tightbinding model with Hamiltonian
where and are real parameters. What is the energy spectrum as a function of the wave vector in the Brillouin zone? How does the energy vary along the boundary of the Brillouin zone between and ? What is the width of the band?
In a real material, each site of the lattice contains an atom with a certain valency. Explain how the conducting properties of the material depend on the valency.
Suppose now that there is a second band, with minimum . For what values of and the valency is the material an insulator?
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Paper 4, Section II, H
2019 comment(a) Let be a real Hilbert space and let be a bilinear map. If is continuous prove that there is an such that for all . [You may use any form of the Banach-Steinhaus theorem as long as you state it clearly.]
(b) Now suppose that defined as above is bilinear and continuous, and assume also that it is coercive: i.e. there is a such that for all . Prove that for any , there exists a unique such that for all .
[Hint: show that there is a bounded invertible linear operator with bounded inverse so that for all . You may use any form of the Riesz representation theorem as long as you state it clearly.]
(c) Define the Sobolev space , where is open and bounded.
(d) Suppose and with , where is the Euclidean norm on . Consider the Dirichlet problem
Using the result of part (b), prove there is a unique weak solution .
(e) Now assume that is the open unit disk in and is a smooth function on . Sketch how you would solve the following variant:
[Hint: Reduce to the result of part (d).]
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Paper 2, Section II, B
2019 commentGive an account of the variational principle for establishing an upper bound on the ground state energy of a Hamiltonian .
A particle of mass moves in one dimension and experiences the potential with an integer. Use a variational argument to prove the virial theorem,
where denotes the expectation value in the true ground state. Deduce that there is no normalisable ground state for .
For the case , use the ansatz to find an estimate for the energy of the ground state.
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Paper 1, Section II, B
2019 commentA particle of mass and charge moving in a uniform magnetic field and electric field is described by the Hamiltonian
where is the canonical momentum.
[ In the following you may use without proof any results concerning the spectrum of the harmonic oscillator as long as they are stated clearly.]
(a) Let . Choose a gauge which preserves translational symmetry in the direction. Determine the spectrum of the system, restricted to states with . The system is further restricted to lie in a rectangle of area , with sides of length and parallel to the - and -axes respectively. Assuming periodic boundary conditions in the -direction, estimate the degeneracy of each Landau level.
(b) Consider the introduction of an additional electric field . Choosing a suitable gauge (with the same choice of vector potential as in part (a)), write down the resulting Hamiltonian. Find the energy spectrum for a particle on again restricted to states with .
Define the group velocity of the electron and show that its -component is given by .
When the system is further restricted to a rectangle of area as above, show that the previous degeneracy of the Landau levels is lifted and determine the resulting energy gap between the ground-state and the first excited state.
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Paper 4, Section II, K
2019 comment(a) Let be such that is finite for any bounded measurable set . State the properties which define a (non-homogeneous) Poisson process on with intensity function .
(b) Let be a Poisson process on with intensity function , and let be a given function. Give a clear statement of the necessary conditions on the pair subject to which is a Poisson process on . When these conditions hold, express the mean measure of in terms of and .
(c) Let be a homogeneous Poisson process on with constant intensity 1 , and let be given by . Show that is a homogeneous Poisson process on with constant intensity .
Let be an increasing sequence of positive random variables such that the points of are Show that has density function
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Paper 3, Section II, K
2019 comment(a) What does it mean to say that a continuous-time Markov chain ) with state space is reversible in equilibrium? State the detailed balance equations, and show that any probability distribution on satisfying them is invariant for the chain.
(b) Customers arrive in a shop in the manner of a Poisson process with rate . There are servers, and capacity for up to people waiting for service. Any customer arriving when the shop is full (in that the total number of customers present is ) is not admitted and never returns. Service times are exponentially distributed with parameter , and they are independent of one another and of the arrivals process. Describe the number of customers in the shop at time as a Markov chain.
Calculate the invariant distribution of , and explain why is the unique invariant distribution. Show that is reversible in equilibrium.
[Any general result from the course may be used without proof, but must be stated clearly.]
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Paper 2, Section II, K
2019 commentLet be a Markov chain on the non-negative integers with generator given by
for a given collection of positive numbers .
(a) State the transition matrix of the jump chain of .
(b) Why is not reversible?
(c) Prove that is transient if and only if .
(d) Assume that . Derive a necessary and sufficient condition on the parameters for to be explosive.
(e) Derive a necessary and sufficient condition on the parameters for the existence of an invariant measure for .
[You may use any general results from the course concerning Markov chains and pure birth processes so long as they are clearly stated.]
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Paper 1, Section II, K
2019 commentLet be a countable set, and let be a Markov transition matrix with for all . Let be a discrete-time Markov chain on the state space with transition matrix .
The continuous-time process is constructed as follows. Let be independent, identically distributed random variables having the exponential distribution with mean 1. Let be a function on such that for all and some constant . Let for . Let and for . Finally, let for .
(a) Explain briefly why is a continuous-time Markov chain on , and write down its generator in terms of and the vector .
(b) What does it mean to say that the chain is irreducible? What does it mean to say a state is (i) recurrent and (ii) positive recurrent?
(c) Show that
(i) is irreducible if and only if is irreducible;
(ii) is recurrent if and only if is recurrent.
(d) Suppose is irreducible and positive recurrent with invariant distribution . Express the invariant distribution of in terms of and .
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Paper 4, Section II, A
2019 commentConsider, for small , the equation
Assume that has bounded solutions with two turning points where and .
(a) Use the WKB approximation to derive the relationship
[You may quote without proof any standard results or formulae from WKB theory.]
(b) In suitable units, the radial Schrödinger equation for a spherically symmetric potential given by , for constant , can be recast in the standard form as:
where and is a small parameter.
Use result to show that the energies of the bound states (i.e are approximated by the expression:
[You may use the result
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Paper 3, Section II, A
2019 comment(a) State Watson's lemma for the case when all the functions and variables involved are real, and use it to calculate the asymptotic approximation as for the integral , where
(b) The Bessel function of the first kind of order has integral representation
where is the Gamma function, and is in general a complex variable. The complex version of Watson's lemma is obtained by replacing with the complex variable , and is valid for and , for some such that . Use this version to derive an asymptotic expansion for as . For what values of is this approximation valid?
[Hint: You may find the substitution useful.]
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Paper 2, Section II, A
2019 comment(a) Define formally what it means for a real valued function to have an asymptotic expansion about , given by
Use this definition to prove the following properties.
(i) If both and have asymptotic expansions about , then also has an asymptotic expansion about
(ii) If has an asymptotic expansion about and is integrable, then
(b) Obtain, with justification, the first three terms in the asymptotic expansion as of the complementary error function, , defined as
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Paper 1, Section I, H
2019 comment(a) State the pumping lemma for context-free languages (CFLs).
(b) Which of the following are CFLs? Justify your answers.
(i) , where is the reverse of the word .
(ii) is a prime .
(iii) and .
(c) Let and be CFLs. Show that the concatenation is also a CFL.
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Paper 4, Section I,
2019 comment(a) Which of the following are regular languages? Justify your answers.
(i) .
(ii) contains an odd number of 's and an even number of 's .
(iii) contains no more than 7 consecutive 0 's .
(b) Consider the language over alphabet defined via
Show that satisfies the pumping lemma for regular languages but is not a regular language itself.
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Paper 3, Section I,
2019 comment(a) Define what it means for a context-free grammar (CFG) to be in Chomsky normal form (CNF). Can a CFG in CNF ever define a language containing ? If denotes the result of converting an arbitrary CFG into one in CNF, state the relationship between and .
(b) Let be a CFG in CNF. Give an algorithm that, on input of any word on the terminals of , decides if or not. Explain why your algorithm works.
(c) Convert the following CFG into a grammar in CNF:
Does in this case? Justify your answer.
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Paper 2, Section I, H
2019 comment(a) Define a recursive set and a recursively enumerable (r.e.) set. Prove that is recursive if and only if both and are r.e. sets.
(b) Let for some fixed and some fixed register machine code . Show that for some fixed register machine code . Hence show that is an r.e. set.
(c) Show that the function defined below is primitive recursive.
[Any use of Church's thesis in your answers should be explicitly stated. In this question denotes the set of non-negative integers.]
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Paper 1, Section II, H
2019 commentLet be a deterministic finite-state automaton (DFA). Define what it means for two states of to be equivalent. Define the minimal DFA for .
Let be a DFA with no inaccessible states, and suppose that is another DFA on the same alphabet as and for which . Show that has at least as many states as . [You may use results from the course as long as you state them clearly.]
Construct a minimal DFA (that is, one with the smallest possible number of states) over the alphabet which accepts precisely the set of binary numbers which are multiples of 7. You may have leading zeros in your inputs (e.g.: 00101). Prove that your DFA is minimal by finding a distinguishing word for each pair of states.
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Paper 3, Section II, 12H
2019 comment(a) State the theorem and the recursion theorem.
(b) State and prove Rice's theorem.
(c) Show that if is partial recursive, then there is some such that
(d) Show there exists some such that has exactly elements.
(e) Given , is it possible to compute whether or not the number of elements of is a (finite) perfect square? Justify your answer.
[In this question denotes the set of non-negative integers. Any use of Church's thesis in your answers should be explicitly stated.]
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Paper 4, Section I, E
2019 comment(a) The angular momentum of a rigid body about its centre of mass is conserved.
Derive Euler's equations,
explaining the meaning of the quantities appearing in the equations.
(b) Show that there are two independent conserved quantities that are quadratic functions of , and give a physical interpretation of them.
(c) Derive a linear approximation to Euler's equations that applies when and . Use this to determine the stability of rotation about each of the three principal axes of an asymmetric top.
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Paper 3, Section I, E
2019 commentA simple harmonic oscillator of mass and spring constant has the equation of motion
(a) Describe the orbits of the system in phase space. State how the action of the oscillator is related to a geometrical property of the orbits in phase space. Derive the action-angle variables and give the form of the Hamiltonian of the oscillator in action-angle variables.
(b) Suppose now that the spring constant varies in time. Under what conditions does the theory of adiabatic invariance apply? Assuming that these conditions hold, identify an adiabatic invariant and determine how the energy and amplitude of the oscillator vary with in this approximation.
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Paper 2, Section I, E
2019 comment(a) State Hamilton's equations for a system with degrees of freedom and Hamilto, where are canonical phase-space variables.
(b) Define the Poisson bracket of two functions and .
(c) State the canonical commutation relations of the variables and .
(d) Show that the time-evolution of any function is given by
(e) Show further that the Poisson bracket of any two conserved quantities is also a conserved quantity.
[You may assume the Jacobi identity,
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Paper 1, Section I, E
2019 comment(a) A mechanical system with degrees of freedom has the Lagrangian , where are the generalized coordinates and .
Suppose that is invariant under the continuous symmetry transformation , where is a real parameter and . State and prove Noether's theorem for this system.
(b) A particle of mass moves in a conservative force field with potential energy , where is the position vector in three-dimensional space.
Let be cylindrical polar coordinates. is said to have helical symmetry if it is of the form
for some constant . Show that a particle moving in a potential with helical symmetry has a conserved quantity that is a linear combination of angular and linear momenta.
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Paper 2, Section II, E
2019 commentThe Lagrangian of a particle of mass and charge moving in an electromagnetic field described by scalar and vector potentials and is
where is the position vector of the particle and .
(a) Show that Lagrange's equations are equivalent to the equation of motion
where
are the electric and magnetic fields.
(b) Show that the related Hamiltonian is
where . Obtain Hamilton's equations for this system.
(c) Verify that the electric and magnetic fields remain unchanged if the scalar and vector potentials are transformed according to
where is a scalar field. Show that the transformed Lagrangian differs from by the total time-derivative of a certain quantity. Why does this leave the form of Lagrange's equations invariant? Show that the transformed Hamiltonian and phase-space variables are related to and by a canonical transformation.
[Hint: In standard notation, the canonical transformation associated with the type-2 generating function is given by
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Paper 4, Section II, E
2019 comment(a) Explain what is meant by a Lagrange top. You may assume that such a top has the Lagrangian
in terms of the Euler angles . State the meaning of the quantities and appearing in this expression.
Explain why the quantity
is conserved, and give two other independent integrals of motion.
Show that steady precession, with a constant value of , is possible if
(b) A rigid body of mass is of uniform density and its surface is defined by
where is a positive constant and are Cartesian coordinates in the body frame.
Calculate the values of and for this symmetric top, when it rotates about the sharp point at the origin of this coordinate system.
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Paper 4, Section I,
2019 comment(a) Describe Diffie-Hellman key exchange. Why is it believed to be a secure system?
(b) Consider the following authentication procedure. Alice chooses public key for the Rabin-Williams cryptosystem. To be sure we are in communication with Alice we send her a 'random item' . On receiving , Alice proceeds to decode using her knowledge of the factorisation of and finds a square root of . She returns to us and we check . Is this authentication procedure secure? Justify your answer.
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Paper 3, Section I, G
2019 commentWhat does it mean to transmit reliably at rate through a binary symmetric channel with error probability ?
Assuming Shannon's second coding theorem (also known as Shannon's noisy coding theorem), compute the supremum of all possible reliable transmission rates of a BSC. Describe qualitatively the behaviour of the capacity as varies. Your answer should address the following cases,
(i) is small,
(ii) ,
(iii) .
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Paper 2, Section I, G
2019 commentDefine the binary Hamming code of length for . Define a perfect code. Show that a binary Hamming code is perfect.
What is the weight of the dual code of a binary Hamming code when
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Paper 1, Section I, G
2019 commentLet and be discrete random variables taking finitely many values. Define the conditional entropy . Suppose is another discrete random variable taking values in a finite alphabet, and prove that
[You may use the equality and the inequality
State and prove Fano's inequality.
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Paper 1, Section II, G
2019 commentWhat does it mean to say that is a binary linear code of length , rank and minimum distance ? Let be such a code.
(a) Prove that .
Let be a codeword with exactly non-zero digits.
(b) Prove that puncturing on the non-zero digits of produces a code of length and minimum distance for some .
(c) Deduce that .
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Paper 2, Section II, G
2019 commentDescribe the Huffman coding scheme and prove that Huffman codes are optimal.
Are the following statements true or false? Justify your answers.
(i) Given messages with probabilities a Huffman coding will assign a unique set of word lengths.
(ii) An optimal code must be Huffman.
(iii) Suppose the words of a Huffman code have word lengths . Then
[Throughout this question you may assume that a decipherable code with prescribed word lengths exists if and only if there is a prefix-free code with the same word lengths.]
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Paper 3, Section I, B
2019 commentConsider a spherically symmetric distribution of mass with density at distance from the centre. Derive the pressure support equation that the pressure has to satisfy for the system to be in static equilibrium.
Assume now that the mass density obeys , for some positive constant A. Determine whether or not the system has a stable solution corresponding to a star of finite radius.
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Paper 4, Section I, B
2019 commentDerive the relation between the neutrino temperature and the photon temperature at a time long after electrons and positrons have become non-relativistic.
[In this question you may work in units of the speed of light, so that . You may also use without derivation the following formulae. The energy density and pressure for a single relativistic species a with a number of degenerate states at temperature are given by
where is Boltzmann's constant, is Planck's constant, and the minus or plus depends on whether the particle is a boson or a fermion respectively. For each species a, the entropy density at temperature is given by,
The effective total number of relativistic species is defined in terms of the numbers of bosonic and fermionic particles in the theory as,
with the specific values for photons, positrons and electrons.]
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Paper 1, Section I, B
2019 comment[You may work in units of the speed of light, so that .]
By considering a spherical distribution of matter with total mass and radius and an infinitesimal mass located somewhere on its surface, derive the Friedmann equation describing the evolution of the scale factor appearing in the relation for a spatially-flat FLRW spacetime.
Consider now a spatially-flat, contracting universe filled by a single component with energy density , which evolves with time as . Solve the Friedmann equation for with .
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Paper 2, Section I, B
2019 comment[You may work in units of the speed of light, so that .]
(a) Combining the Friedmann and continuity equations
derive the Raychaudhuri equation (also known as the acceleration equation) which expresses in terms of the energy density and the pressure .
(b) Assuming an equation of state with constant , for what is the expansion of the universe accelerated or decelerated?
(c) Consider an expanding, spatially-flat FLRW universe with both a cosmological constant and non-relativistic matter (also known as dust) with energy densities and respectively. At some time corresponding to , the energy densities of these two components are equal . Is the expansion of the universe accelerated or decelerated at this time?
(d) For what numerical value of does the universe transition from deceleration to acceleration?
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Paper 3, Section II, B
2019 comment[You may work in units of the speed of light, so that ]
Consider the process where protons and electrons combine to form neutral hydrogen atoms;
Let and denote the number densities for protons, electrons and hydrogen atoms respectively. The ionization energy of hydrogen is denoted . State and derive 's equation for the ratio , clearly describing the steps required.
[You may use without proof the following formula for the equilibrium number density of a non-relativistic species with degenerate states of mass at temperature such that ,
where is the chemical potential and and are the Boltzmann and Planck constants respectively.]
The photon number density is given as
where . Consider now the fractional ionization . In our universe where is the baryon-to-photon number ratio. Find an expression for the ratio
in terms of and the particle masses. One might expect neutral hydrogen to form at a temperature given by , but instead in our universe it forms at the much lower temperature . Briefly explain why this happens. Estimate the temperature at which neutral hydrogen would form in a hypothetical universe with . Briefly explain your answer.
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Paper 1, Section II, 15B
2019 comment[You may work in units of the speed of light, so that .]
Consider a spatially-flat FLRW universe with a single, canonical, homogeneous scalar field with a potential . Recall the Friedmann equation and the Raychaudhuri equation (also known as the acceleration equation)
(a) Assuming , derive the equations of motion for , i.e.
(b) Assuming the special case , find , for some initial value in the slow-roll approximation, i.e. assuming that and .
(c) The number of efoldings is defined by . Using the chain rule, express first in terms of and then in terms of . Write the resulting relation between and in terms of and only, using the slow-roll approximation.
(d) Compute the number of efoldings of expansion between some initial value and a final value (so that throughout).
(e) Discuss qualitatively the horizon and flatness problems in the old hot big bang model (i.e. without inflation) and how inflation addresses them.
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Paper 4, Section II, H
2019 comment(a) Let be a regular curve without self-intersection given by with for and let be the surface of revolution defined globally by the parametrisation
where , i.e. . Compute its mean curvature and its Gaussian curvature .
(b) Define what it means for a regular surface to be minimal. Give an example of a minimal surface which is not locally isometric to a cone, cylinder or plane. Justify your answer.
(c) Let be a regular surface such that . Is it necessarily the case that given any , there exists an open neighbourhood of such that lies on some sphere in ? Justify your answer.
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Paper 3, Section II, H
2019 comment(a) Let be a regular curve without self intersection given by with for .
Consider the local parametrisation given by
where .
(i) Show that the image defines a regular surface in .
(ii) If is a geodesic in parametrised by arc length, then show that is constant in . If denotes the angle that the geodesic makes with the parallel , then show that is constant in .
(b) Now assume that extends to a smooth curve such that . Let be the closure of in .
(i) State a necessary and sufficient condition on for to be a compact regular surface. Justify your answer.
(ii) If is a compact regular surface, and is a geodesic, show that there exists a non-empty open subset such that .
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Paper 2, Section II, H
2019 comment(a) Let be a smooth regular curve parametrised by arclength. For , define the curvature and (where defined) the torsion of . What condition must be satisfied in order for the torsion to be defined? Derive the Frenet equations.
(b) If is defined and equal to 0 for all , show that lies in a plane.
(c) State the fundamental theorem for regular curves in , giving necessary and sufficient conditions for when curves and are related by a proper Euclidean motion.
(d) Now suppose that is another smooth regular curve parametrised by arclength, and that and are its curvature and torsion. Determine whether the following statements are true or false. Justify your answer in each case.
(i) If whenever it is defined, then lies in a plane.
(ii) If is defined and equal to 0 for all but one value of in , then lies in a plane.
(iii) If for all and are defined for all , and for all , then and are related by a rigid motion.
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Paper 1, Section II, H
2019 commentLet be an integer.
(a) Show that defines a submanifold of and identify explicitly its tangent space for any .
(b) Show that the matrix group defines a submanifold. Identify explicitly the tangent space for any .
(c) Given , show that the set defines a submanifold and compute its dimension. For , is it ever the case that and are transversal?
[You may use standard theorems from the course concerning regular values and transversality.]
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Paper 4, Section II, E
2019 commentConsider the dynamical system
for .
Find all fixed points of this system. Find the three different values of at which bifurcations appear. For each such value give the location of all bifurcations. For each of these, what types of bifurcation are suggested from this analysis?
Use centre manifold theory to analyse these bifurcations. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.
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Paper 3, Section II, E
2019 commentConsider a dynamical system of the form
on and , where and are real constants and .
(a) For , by considering a function of the form , show that all trajectories in are either periodic orbits or a fixed point.
(b) Using the same , show that no periodic orbits in persist for small and if .
[Hint: for on the periodic orbits with period , show that and hence that .]
(c) By considering Dulac's criterion with , show that there are no periodic orbits in if .
(d) Purely by consideration of the existence of fixed points in and their Poincaré indices, determine those for which the possibility of periodic orbits can be excluded.
(e) Combining the results above, sketch the plane showing where periodic orbits in might still be possible.
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Paper 2, Section II, E
2019 commentFor a map give the definitions of chaos according to (i) Devaney (Dchaos) and (ii) Glendinning (G-chaos).
Consider the dynamical system
on , for (note that is not necessarily an integer). For both definitions of chaos, show that this system is chaotic.
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Paper 1, Section II, E
2019 commentFor a dynamical system of the form , give the definition of the alpha-limit set and the omega-limit set of a point .
Consider the dynamical system
where and is a real constant. Answer the following for all values of , taking care over boundary cases (both in and in ).
(i) What symmetries does this system have?
(ii) Find and classify the fixed points of this system.
(iii) Does this system have any periodic orbits?
(iv) Give and (considering all ).
(v) For , give the orbit of (considering all ). You should give your answer in the form , and specify the range of .
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Paper 4, Section II, E
2019 commentConsider a medium in which the electric displacement and magnetising field are linearly related to the electric and magnetic fields respectively with corresponding polarisation constants and ;
Write down Maxwell's equations for and in the absence of free charges and currents.
Consider EM waves of the form,
Find conditions on the electric and magnetic polarisation vectors and , wave-vector and angular frequency such that these fields satisfy Maxwell's equations for the medium described above. At what speed do the waves propagate?
Consider two media, filling the regions and in three dimensional space, and having two different values and of the electric polarisation constant. Suppose an electromagnetic wave is incident from the region resulting in a transmitted wave in the region and also a reflected wave for . The angles of incidence, reflection and transmission are denoted and respectively. By constructing a corresponding solution of Maxwell's equations, derive the law of reflection and Snell's law of refraction, where are the indices of refraction of the two media.
Consider the special case in which the electric polarisation vectors and of the incident, reflected and transmitted waves are all normal to the plane of incidence (i.e. the plane containing the corresponding wave-vectors). By imposing appropriate boundary conditions for and at , show that,
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Paper 3, Section II, E
2019 commentA time-dependent charge distribution localised in some region of size near the origin varies periodically in time with characteristic angular frequency . Explain briefly the circumstances under which the dipole approximation for the fields sourced by the charge distribution is valid.
Far from the origin, for , the vector potential sourced by the charge distribution is given by the approximate expression
where is the corresponding current density. Show that, in the dipole approximation, the large-distance behaviour of the magnetic field is given by,
where is the electric dipole moment of the charge distribution. Assuming that, in the same approximation, the corresponding electric field is given as , evaluate the flux of energy through the surface element of a large sphere of radius centred at the origin. Hence show that the total power radiated by the charge distribution is given by
A particle of charge and mass undergoes simple harmonic motion in the -direction with time period and amplitude such that
Here is a unit vector in the -direction. Calculate the total power radiated through a large sphere centred at the origin in the dipole approximation and determine its time averaged value,
For what values of the parameters and is the dipole approximation valid?
Now suppose that the energy of the particle with trajectory is given by the usual non-relativistic formula for a harmonic oscillator i.e. , and that the particle loses energy due to the emission of radiation at a rate corresponding to the time-averaged power . Work out the half-life of this system (i.e. the time such that . Explain why the non-relativistic approximation for the motion of the particle is reliable as long as the dipole approximation is valid.
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Paper 1, Section II, E
2019 commentA relativistic particle of charge and mass moves in a background electromagnetic field. The four-velocity of the particle at proper time is determined by the equation of motion,
Here , where is the electromagnetic field strength tensor and Lorentz indices are raised and lowered with the metric tensor . In the case of a constant, homogeneous field, write down the solution of this equation giving in terms of its initial value .
[In the following you may use the relation, given below, between the components of the field strength tensor , for , and those of the electric and magnetic fields and ,
for
Suppose that, in some inertial frame with spacetime coordinates and , the electric and magnetic fields are parallel to the -axis with magnitudes and respectively. At time the 3 -velocity of the particle has initial value . Find the subsequent trajectory of the particle in this frame, giving coordinates and as functions of the proper time .
Find the motion in the -direction explicitly, giving as a function of coordinate time . Comment on the form of the solution at early and late times. Show that, when projected onto the plane, the particle undergoes circular motion which is periodic in proper time. Find the radius of the circle and proper time period of the motion in terms of and . The resulting trajectory therefore has the form of a helix with varying pitch where is the distance in the -direction travelled by the particle during the 'th period of its motion in the plane. Show that, for ,
where is a constant which you should determine.
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Paper 4, Section II, A
2019 comment(a) Show that the Stokes flow around a rigid moving sphere has the minimum viscous dissipation rate of all incompressible flows which satisfy the no-slip boundary conditions on the sphere.
(b) Let , where and are solutions of Laplace's equation, i.e. and .
(i) Show that is incompressible.
(ii) Show that satisfies Stokes equation if the pressure .
(c) Consider a rigid sphere moving with velocity . The Stokes flow around the sphere is given by
where the origin is chosen to be at the centre of the sphere. Find the values for and which ensure no-slip conditions are satisfied on the sphere.
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Paper 2, Section II, A
2019 commentA viscous fluid is contained in a channel between rigid planes and . The fluid in the upper region (with ) has dynamic viscosity while the fluid in the lower region has dynamic viscosity . The plane at moves with velocity and the plane at moves with velocity , both in the direction. You may ignore the effect of gravity.
(a) Find the steady, unidirectional solution of the Navier-Stokes equations in which the interface between the two fluids remains at .
(b) Using the solution from part (a):
(i) calculate the stress exerted by the fluids on the two boundaries;
(ii) calculate the total viscous dissipation rate in the fluids;
(iii) demonstrate that the rate of working by boundaries balances the viscous dissipation rate in the fluids.
(c) Consider the situation where . Defining the volume flux in the upper region as and the volume flux in the lower region as , show that their ratio is independent of and satisfies
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Paper 3, Section II, A
2019 commentFor a fluid with kinematic viscosity , the steady axisymmetric boundary-layer equations for flow primarily in the -direction are
where is the fluid velocity in the -direction and is the fluid velocity in the -direction. A thin, steady, axisymmetric jet emerges from a point at the origin and flows along the -axis in a fluid which is at rest far from the -axis.
(a) Show that the momentum flux
is independent of the position along the jet. Deduce that the thickness of the jet increases linearly with . Determine the scaling dependence on of the centre-line velocity . Hence show that the jet entrains fluid.
(b) A similarity solution for the streamfunction,
exists if satisfies the second order differential equation
Using appropriate boundary and normalisation conditions (which you should state clearly) to solve this equation, show that
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Paper 1, Section II, A
2019 commentA disc of radius and weight hovers at a height on a cushion of air above a horizontal air table - a fine porous plate through which air of density and dynamic viscosity is pumped upward at constant speed . You may assume that the air flow is axisymmetric with no flow in the azimuthal direction, and that the effect of gravity on the air may be ignored.
(a) Write down the relevant components of the Navier-Stokes equations. By estimating the size of the individual terms, simplify these equations when and .
(b) Explain briefly why it is reasonable to expect that the vertical velocity of the air below the disc is a function of distance above the air table alone, and thus find the steady pressure distribution below the disc. Hence show that
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Paper 4, Section I, A
2019 commentA single-valued function can be defined, for , by means of an integral as:
(a) Choose a suitable branch-cut with the integrand taking a value at the origin on the upper side of the cut, i.e. at , and describe suitable paths of integration in the two cases and .
(b) Construct the multivalued function by analytic continuation.
(c) Express arcsin in terms of and deduce the periodicity property of .
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Paper 3, Section I, A
2019 commentThe equation
has solutions of the form
for suitably chosen contours and some suitable function .
(a) Find and determine the required condition on , which you should express in terms of and .
(b) Use the result of part (a) to specify a possible contour with the help of a clearly labelled diagram.
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Paper 2, Section I, A
2019 commentAssume that as and that is analytic in the upper half-plane (including the real axis). Evaluate
where is a positive real number.
[You must state clearly any standard results involving contour integrals that you use.]
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Paper 1, Section I, A
2019 commentThe Beta function is defined by
where , and is the Gamma function.
(a) By using a suitable substitution and properties of Beta and Gamma functions, show that
(b) Deduce that
where is the complete elliptic integral, defined as
[Hint: You might find the change of variable helpful in part (b).]
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Paper 2, Section II, A
2019 commentThe Riemann zeta function is defined as
for , and by analytic continuation to the rest of except at singular points. The integral representation of ( ) for is given by
where is the Gamma function.
(a) The Hankel representation is defined as
Explain briefly why this representation gives an analytic continuation of as defined in ( ) to all other than , using a diagram to illustrate what is meant by the upper limit of the integral in .
[You may assume .]
(b) Find
where is an integer and the poles are simple.
(c) By considering
where is a suitably modified Hankel contour and using the result of part (b), derive the reflection formula:
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Paper 1, Section II, A
2019 comment(a) Consider the Papperitz symbol (or P-symbol):
Explain in general terms what this -symbol represents.
[You need not write down any differential equations explicitly, but should provide an explanation of the meaning of and
(b) Prove that the action of on results in the exponential shifting,
[Hint: It may prove useful to start by considering the relationship between two solutions, and , which satisfy the -equations described by the respective -symbols ( ) and ( -]
(c) Explain what is meant by a Möbius transformation of a second order differential equation. By using suitable transformations acting on , show how to obtain the symbol
which corresponds to the hypergeometric equation.
(d) The hypergeometric function is defined to be the solution of the differential equation corresponding to that is analytic at with , which corresponds to the exponent zero. Use exponential shifting to show that the second solution, which corresponds to the exponent , is
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Paper 1, Section II, 18F
2019 comment(a) Suppose are fields and are distinct embeddings of into . Prove that there do not exist elements of (not all zero) such that
(b) For a finite field extension of a field and for distinct automorphisms of , show that . In particular, if is a finite group of field automorphisms of a field with the fixed field, deduce that .
(c) If with independent transcendentals over , consider the group generated by automorphisms and of , where
Prove that and that .
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Paper 2, Section II, F
2019 commentFor any prime , explain briefly why the Galois group of over is cyclic of order , where if if , and if
Show that the splitting field of over is an extension of degree 20 .
For any prime , prove that does not have an irreducible cubic as a factor. For or , show that is the product of a linear factor and an irreducible quartic over . For , show that either is irreducible over or it splits completely.
[You may assume the reduction mod p criterion for finding cycle types in the Galois group of a monic polynomial over and standard facts about finite fields.]
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Paper 3, Section II, F
2019 commentLet be a field. For a positive integer, consider , where either char , or char with not dividing ; explain why the polynomial has distinct roots in a splitting field.
For a positive integer, define the th cyclotomic polynomial and show that it is a monic polynomial in . Prove that is irreducible over for all . [Hint: If , with and monic irreducible with , and is a root of , show first that is a root of for any prime not dividing .]
Let ; by considering the product , or otherwise, show that is irreducible over .
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Paper 4, Section II,
2019 commentState (without proof) a result concerning uniqueness of splitting fields of a polynomial.
Given a polynomial with distinct roots, what is meant by its Galois group ? Show that is irreducible over if and only if acts transitively on the roots of .
Now consider an irreducible quartic of the form . If denotes a root of , show that the splitting field is . Give an explicit description of in the cases:
(i) , and
(ii) .
If is a square in , deduce that . Conversely, if Gal , show that is invariant under at least two elements of order two in the Galois group, and deduce that is a square in .
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Paper 4, Section II, D
2019 comment(a) Consider the spherically symmetric spacetime metric
where and are functions of and . Use the Euler-Lagrange equations for the geodesics of the spacetime to compute all non-vanishing Christoffel symbols for this metric.
(b) Consider the static limit of the line element where and are functions of the radius only, and let the matter coupled to gravity be a spherically symmetric fluid with energy momentum tensor
where the pressure and energy density are also functions of the radius . For these Tolman-Oppenheimer-Volkoff stellar models, the Einstein and matter equations and reduce to
Consider now a constant density solution to the above Einstein and matter equations, where takes the non-zero constant value out to a radius and for . Show that for such a star,
and that the pressure at the centre of the star is
Show that diverges if [Hint: at the surface of the star the pressure vanishes:
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Paper 2, Section II, D
2019 commentConsider the spacetime metric
where and are constants.
(a) Write down the Lagrangian for geodesics in this spacetime, determine three independent constants of motion and show that geodesics obey the equation
where is constant, the overdot denotes differentiation with respect to an affine parameter and is a potential function to be determined.
(b) Sketch the potential for the case of null geodesics, find any circular null geodesics of this spacetime, and determine whether they are stable or unstable.
(c) Show that has two positive roots and if and that these satisfy the relation .
(d) Describe in one sentence the physical significance of those points where .
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Paper 3, Section II, D
2019 comment(a) Let be a manifold with coordinates . The commutator of two vector fields and is defined as
(i) Show that transforms like a vector field under a change of coordinates from to .
(ii) Show that the commutator of any two basis vectors vanishes, i.e.
(iii) Show that if and are linear combinations (not necessarily with constant coefficients) of vector fields that all commute with one another, then the commutator is a linear combination of the same fields .
[You may use without proof the following relations which hold for any vector fields and any function :
but you should clearly indicate each time relation , or (3) is used.]
(b) Consider the 2-dimensional manifold with Cartesian coordinates carrying the Euclidean metric .
(i) Express the coordinate basis vectors and , where and denote the usual polar coordinates, in terms of their Cartesian counterparts.
(ii) Define the unit vectors
and show that are not a coordinate basis, i.e. there exist no coordinates such that and .
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Paper 1, Section II, D
2019 commentLet be a spacetime and the Levi-Civita connection of the metric . The Riemann tensor of this spacetime is given in terms of the connection by
The contracted Bianchi identities ensure that the Einstein tensor satisfies
(a) Show that the Riemann tensor obeys the symmetry
(b) Show that a vector field satisfies the Ricci identity
Calculate the analogous expression for a rank tensor , i.e. calculate in terms of the Riemann tensor.
(c) Let be a vector that satisfies the Killing equation
Use the symmetry relation of part (a) to show that
where is the Ricci tensor.
(d) Show that
and use the result of part (b) to show that the right hand side evaluates to zero, hence showing that .
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Paper 4, Section II, G
2019 commentState and prove Hall's theorem.
Let be an even positive integer. Let be the power set of . For , let . Let be the graph with vertex set where are adjacent if and only if . [Here, denotes the symmetric difference of and , given by
Let . Why is the induced subgraph bipartite? Show that it contains a matching from to .
A chain in is a subset such that whenever we have or . What is the least positive integer such that can be partitioned into pairwise disjoint chains? Justify your answer.
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Paper 3, Section II, G
2019 comment(a) What does it mean to say that a graph is bipartite?
(b) Show that is bipartite if and only if it contains no cycles of odd length.
(c) Show that if is bipartite then
as .
[You may use without proof the Erdós-Stone theorem provided it is stated precisely.]
(d) Let be a graph of order with edges. Let be a random subset of containing each vertex of independently with probability . Let be the number of edges with precisely one vertex in . Find, with justification, , and deduce that contains a bipartite subgraph with at least edges.
By using another method of choosing a random subset of , or otherwise, show that if is even then contains a bipartite subgraph with at least edges.
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Paper 2, Section II, 17G
2019 comment(a) Suppose that the edges of the complete graph are coloured blue and yellow. Show that it must contain a monochromatic triangle. Does this remain true if is replaced by ?
(b) Let . Suppose that the edges of the complete graph are coloured blue and yellow. Show that it must contain edges of the same colour with no two sharing a vertex. Is there any for which this remains true if is replaced by ?
(c) Now let . Suppose that the edges of the complete graph are coloured blue and yellow in such a way that there are a blue triangle and a yellow triangle with no vertices in common. Show that there are also a blue triangle and a yellow triangle that do have a vertex in common. Hence, or otherwise, show that whenever the edges of the complete graph are coloured blue and yellow it must contain monochromatic triangles, all of the same colour, with no two sharing a vertex. Is there any for which this remains true if is replaced by ? [You may assume that whenever the edges of the complete graph are coloured blue and yellow it must contain two monochromatic triangles of the same colour with no vertices in common.]
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Paper 1, Section II, 17G
2019 commentLet be a connected -regular graph.
(a) Show that is an eigenvalue of with multiplicity 1 and eigenvector
(b) Suppose that is strongly regular. Show that has at most three distinct eigenvalues.
(c) Conversely, suppose that has precisely three distinct eigenvalues and . Let be the adjacency matrix of and let
Show that if is an eigenvector of that is not a scalar multiple of then . Deduce that is a scalar multiple of the matrix whose entries are all equal to one. Hence show that, for depends only on whether or not vertices and are adjacent, and so is strongly regular.
(d) Which connected -regular graphs have precisely two eigenvalues? Justify your answer.
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Paper 3, Section II, C
2019 commentSuppose is a smooth one-parameter group of transformations acting on , with infinitesimal generator
(a) Define the prolongation of , and show that
where you should give an explicit formula to determine the recursively in terms of and .
(b) Find the prolongation of each of the following generators:
(c) Given a smooth, real-valued, function , the Schwarzian derivative is defined by,
Show that,
for where are real functions which you should determine. What can you deduce about the symmetries of the equations: (i) , (ii) , (iii) ?
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Paper 2, Section II, C
2019 commentSuppose is a smooth, real-valued, function of which satisfies for all and as . Consider the Sturm-Liouville operator:
which acts on smooth, complex-valued, functions . You may assume that for any there exists a unique function which satisfies:
and has the asymptotic behaviour:
(a) By analogy with the standard Schrödinger scattering problem, define the reflection and transmission coefficients: . Show that . [Hint: You may wish to consider for suitable functions and
(b) Show that, if , there exists no non-trivial normalizable solution to the equation
Assume now that , such that and 0 as . You are given that the operator defined by:
satisfies:
(c) Show that form a Lax pair if the Harry Dym equation,
is satisfied. [You may assume .]
(d) Assuming that solves the Harry Dym equation, find how the transmission and reflection amplitudes evolve as functions of .
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Paper 1, Section II, C
2019 commentLet be equipped with its standard Poisson bracket.
(a) Given a Hamiltonian function , write down Hamilton's equations for . Define a first integral of the system and state what it means that the system is integrable.
(b) Show that if then every Hamiltonian system is integrable whenever
Let be another phase space, equipped with its standard Poisson bracket. Suppose that is a Hamiltonian function for . Define and let the combined phase space be equipped with the standard Poisson bracket.
(c) Show that if and are both integrable, then so is , where the combined Hamiltonian is given by:
(d) Consider the -dimensional simple harmonic oscillator with phase space and Hamiltonian given by:
where . Using the results above, or otherwise, show that is integrable for .
(e) Is it true that every bounded orbit of an integrable system is necessarily periodic? You should justify your answer.
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Paper 3, Section II, H
2019 comment(a) Let be a Banach space and consider the open unit ball . Let be a bounded operator. Prove that .
(b) Let be the vector space of all polynomials in one variable with real coefficients. Let be any norm on . Show that is not complete.
(c) Let be entire, and assume that for every there is such that where is the -th derivative of . Prove that is a polynomial.
[You may use that an entire function vanishing on an open subset of must vanish everywhere.]
(d) A Banach space is said to be uniformly convex if for every there is such that for all such that and , one has . Prove that is uniformly convex.
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Paper 4, Section II, H
2019 comment(a) State and prove the Riesz representation theorem for a real Hilbert space .
[You may use that if is a real Hilbert space and is a closed subspace, then
(b) Let be a real Hilbert space and a bounded linear operator. Show that is invertible if and only if both and are bounded below. [Recall that an operator is bounded below if there is such that for all .]
(c) Consider the complex Hilbert space of two-sided sequences,
with norm . Define by . Show that is unitary and find the point spectrum and the approximate point spectrum of .
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Paper 2, Section II, H
2019 comment(a) State the real version of the Stone-Weierstrass theorem and state the UrysohnTietze extension theorem.
(b) In this part, you may assume that there is a sequence of polynomials such that as .
Let be a continuous piecewise linear function which is linear on and on . Using the polynomials mentioned above (but not assuming any form of the Stone-Weierstrass theorem), prove that there are polynomials such that as .
(d) Which of the following families of functions are relatively compact in with the supremum norm? Justify your answer.
[In this question denotes the set of positive integers.]
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Paper 1, Section II, H
2019 commentLet be the space of real-valued sequences with only finitely many nonzero terms.
(a) For any , show that is dense in . Is dense in Justify your answer.
(b) Let , and let be an operator that is bounded in the -norm, i.e., there exists a such that for all . Show that there is a unique bounded operator satisfying , and that .
(c) For each and for each determine if there is a bounded operator from to (in the norm) whose restriction to is given by :
(d) Let be a normed vector space such that the closed unit ball is compact. Prove that is finite dimensional.
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Paper 4, Section II, I
2019 commentDefine the cardinals , and explain briefly why every infinite set has cardinality
Show that if is an infinite cardinal then .
Let be infinite sets. Show that must have the same cardinality as for some .
Let be infinite sets, no two of the same cardinality. Is it possible that has the same cardinality as some ? Justify your answer.
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Paper 3, Section II, I
2019 commentDefine the von Neumann hierarchy of sets . Show that each is transitive, and explain why whenever . Prove that every set is a member of some .
Which of the following are true and which are false? Give proofs or counterexamples as appropriate. [You may assume standard properties of rank.]
(i) If the rank of a set is a (non-zero) limit then is infinite.
(ii) If the rank of a set is countable then is countable.
(iii) If every finite subset of a set has rank at most then has rank at most .
(iv) For every ordinal there exists a set of .
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Paper 2, Section II, I
2019 commentGive the inductive and synthetic definitions of ordinal addition, and prove that they are equivalent.
Which of the following assertions about ordinals and are always true, and which can be false? Give proofs or counterexamples as appropriate.
(i) .
(ii) If and are uncountable then .
(iii) .
(iv) If and are infinite and then .
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Paper 1, Section II, I
2019 commentState the completeness theorem for propositional logic. Explain briefly how the proof of this theorem changes from the usual proof in the case when the set of primitive propositions may be uncountable.
State the compactness theorem and the decidability theorem, and deduce them from the completeness theorem.
A poset is called two-dimensional if there exist total orders and on such that if and only if and . By applying the compactness theorem for propositional logic, show that if every finite subset of a poset is two-dimensional then so is the poset itself.
[Hint: Take primitive propositions and , for each distinct , with the intended interpretation that is true if and only if and is true if and only if
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Paper 4, Section I, C
2019 comment(a) A variant of the classic logistic population model is given by:
where .
Show that for small , the constant solution is stable.
Allow to increase. Express in terms of the value of at which the constant solution loses stability.
(b) Another variant of the logistic model is given by this equation:
where . When is the constant solution stable for this model?
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Paper 3, Section I,
2019 commentA model of wound healing in one spatial dimension is given by
where gives the density of healthy tissue at spatial position at time and and are positive constants.
By setting where , seek a steady travelling wave solution where tends to one for large negative and tends to zero for large positive . By linearising around the leading edge, where , find the possible wave speeds of the system. Assuming that the full nonlinear system will settle to the slowest possible speed, express the wave speed as a function of and .
Consider now a situation where the tissue is destroyed in some window of length , i.e. for for some constant and is equal to one elsewhere. Explain what will happen for subsequent times, illustrating your answer with sketches of . Determine approximately how long it will take for this wound to heal (in the sense that is close to one everywhere).
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Paper 2, Section I, C
2019 commentAn activator-inhibitor system for and is described by the equations
where .
Find the range of for which the spatially homogeneous system has a stable equilibrium solution with and .
For the case when the homogeneous system is stable, consider spatial perturbations proportional to to the equilibrium solution found above. Give a condition on in terms of for the system to have a Turing instability (a spatial instability).
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Paper 1, Section ,
2019 commentAn animal population has annual dynamics, breeding in the summer and hibernating through the winter. At year , the number of individuals alive who were born a years ago is given by . Each individual of age gives birth to offspring, and after the summer has a probability of dying during the winter. [You may assume that individuals do not give birth during the year in which they are born.]
Explain carefully why the following equations, together with initial conditions, are appropriate to describe the system:
Seek a solution of the form where and , for , are constants. Show must satisfy where
Explain why, for any reasonable set of parameters and , the equation has a unique solution. Explain also how can be used to determine if the population will grow or shrink.
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Paper 3, Section II, C
2019 comment(a) A stochastic birth-death process has a master equation given by
where is the probability that there are individuals in the population at time for and for .
(i) Give a brief interpretation of and .
(ii) Derive an equation for , where is the generating function
(iii) Assuming that the generating function takes the form
find and hence show that, as , both the mean and variance of the population size tend to constant values, which you should determine.
(b) Now suppose an extra process is included: individuals are added to the population at rate .
(i) Write down the new master equation, and explain why, for , the approach used in part (a) will fail.
(ii) By working with the master equation directly, find a differential equation for the rate of change of the mean population size .
(iii) Now take for positive constants and . Show that for the mean population size tends to a constant, which you should determine. Briefly describe what happens for .
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Paper 4, Section II, C
2019 commentA model of an infectious disease in a plant population is given by
where is the density of healthy plants and is the density of diseased plants at time and is a positive constant.
(a) Give an interpretation of what each of the terms in equations (1) and (2) represents in terms of the dynamics of the plants. What does the coefficient represent? What can you deduce from the equations about the effect of the disease on the plants?
(b) By finding all fixed points for and and analysing their stability, explain what will happen to a healthy plant population if the disease is introduced. Sketch the phase diagram, treating the cases and separately.
(c) Define new variables for the total plant population density and for the proportion of the population that is diseased. Starting from equations (1) and (2) above, derive equations for and purely in terms of and . Without carrying out a full fixed point analysis, explain how this system can be used directly to show the same results you had in part (b). [Hint: start by considering the dynamics of alone.]
(d) Suppose now that in an attempt to control disease, plants are culled at a rate per capita, independently of whether the plants are healthy or diseased. Write down the modified versions of equations (1) and (2). Use these to build updated equations for and . Without carrying out a detailed fixed point analysis, what can you deduce about the effect of culling? Give the range of for which culling can effectively control the disease.
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Paper 4, Section II, 20G
2019 comment(a) Let be a number field, and suppose there exists such that . Let denote the minimal polynomial of , and let be a prime. Let denote the reduction modulo of , and let
denote the factorisation of in as a product of powers of distinct monic irreducible polynomials , where are all positive integers.
For each , let be any polynomial with reduction modulo equal to , and let . Show that are distinct, non-zero prime ideals of , and that there is a factorisation
and that .
(b) Let be a number field of degree , and let be a prime. Suppose that there is a factorisation
where are distinct, non-zero prime ideals of with for each . Use the result of part (a) to show that if then there is no such that .
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Paper 2, Section II, G
2019 comment(a) Let be a number field. State Minkowski's upper bound for the norm of a representative for a given class of the ideal class group .
(b) Now let and . Using Dedekind's criterion, or otherwise, factorise the ideals and as products of non-zero prime ideals of .
(c) Show that is cyclic, and determine its order.
[You may assume that
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Paper 1, Section II, 20G
2019 commentLet .
(a) Write down the ring of integers .
(b) State Dirichlet's unit theorem, and use it to determine all elements of the group of units .
(c) Let denote the ideal generated by . Show that the group
is cyclic, and find a generator.
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Paper 4, Section I, I
2019 commentShow that the product
and the series
are both divergent.
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Paper 3, Section I, I
2019 commentLet be a positive definite binary quadratic form with integer coefficients. What does it mean to say that is reduced? Show that if is reduced and has discriminant , then and . Deduce that for fixed , there are only finitely many reduced of discriminant .
Find all reduced positive definite binary quadratic forms of discriminant .
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Paper 2, Section I, I
2019 commentDefine the Jacobi symbol , where and is odd and positive.
State and prove the Law of Quadratic Reciprocity for the Jacobi symbol. [You may use Quadratic Reciprocity for the Legendre symbol without proof but should state it clearly.]
Compute the Jacobi symbol .
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Paper 1, Section I, I
2019 comment(a) State and prove the Chinese remainder theorem.
(b) Let be an odd positive composite integer, and a positive integer with . What does it mean to say that is a Fermat pseudoprime to base b? Show that 35 is a Fermat pseudoprime to base if and only if is congruent to one of or .
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Paper 4, Section II, I
2019 comment(a) Let be positive integers, and a positive real number. Show that for every , if , then , where , are sequences of integers satisfying
Show that , and that lies between and .
(b) Show that if is the continued fraction expansion of a positive irrational , then as .
(c) Let the convergents of the continued fraction be . Using part (a) or otherwise, show that the -th and -th convergents of are and respectively.
(d) Show that if is a purely periodic continued fraction with convergents , then , where . Deduce that if is the other root of , then .
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Paper 3, Section II, I
2019 commentLet be a prime.
(a) What does it mean to say that an integer is a primitive root ?
(b) Let be an integer with . Let
Show that . [Recall that by convention .]
(c) Let for some , and let . Show that for any or , and that
Hence show that there exist integers , not all divisible by , such that .
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Paper 4, Section II, C
2019 commentFor a 2-periodic analytic function , its Fourier expansion is given by the formula
(a) Consider the two-point boundary value problem
with periodic boundary conditions . Construct explicitly the infinite dimensional linear algebraic system that arises from the application of the Fourier spectral method to the above equation, and explain how to truncate the system to a finitedimensional one.
(b) A rectangle rule is applied to computing the integral of a 2-periodic analytic function :
Find an expression for the error of , in terms of , and show that has a spectral rate of decay as . [In the last part, you may quote a relevant theorem about .]
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Paper 2, Section II, C
2019 commentThe Poisson equation on the unit square, equipped with zero boundary conditions, is discretized with the 9-point scheme:
where , and are the grid points with . We also assume that .
(a) Prove that all tridiagonal symmetric Toeplitz (TST-) matrices
share the same eigenvectors with the components for . Find expressions for the corresponding eigenvalues for . Deduce that , where and is the matrix
(b) Show that, by arranging the grid points ( into a one-dimensional array by columns, the 9 -points scheme results in the following system of linear equations of the form
where , the vectors are portions of , respectively, and are TST-matrices whose elements you should determine.
(c) Using , show that (2) is equivalent to
where and are diagonal matrices.
(d) Show that, by appropriate reordering of the grid, the system (3) is reduced to uncoupled systems of the form
Determine the elements of the matrices .
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Paper 3, Section II, 40C
2019 commentThe diffusion equation
with the initial condition , and boundary conditions , is discretised by with . The Courant number is given by .
(a) The system is solved numerically by the method
Prove directly that implies convergence.
(b) Now consider the method
where and are real constants. Using an eigenvalue analysis and carefully justifying each step, determine conditions on and for this method to be stable.
[You may use the notation for the tridiagonal matrix with along the diagonal, and along the sub-and super-diagonals and use without proof any relevant theorems about such matrices.]
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Paper 1, Section II, C
2019 comment(a) Describe the Jacobi method for solving a system of linear equations as a particular case of splitting, and state the criterion for its convergence in terms of the iteration matrix.
(b) For the case when
find the exact range of the parameter for which the Jacobi method converges.
(c) State the Householder-John theorem and deduce that the Jacobi method converges if is a symmetric positive-definite tridiagonal matrix.
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Paper 4, Section II, B
2019 commentDefine the spin raising and spin lowering operators and . Show that
where and .
Two spin- particles, with spin operators and , have a Hamiltonian
where and are constants. Express in terms of the two particles' spin raising and spin lowering operators and the corresponding -components , . Hence find the eigenvalues of . Show that there is a unique groundstate in the limit and that the first excited state is triply degenerate in this limit. Explain this degeneracy by considering the action of the combined spin operator on the energy eigenstates.
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Paper 3, Section II, B
2019 commentConsider the Hamiltonian , where is a small perturbation. If , write down an expression for the eigenvalues of , correct to second order in the perturbation, assuming the energy levels of are non-degenerate.
In a certain three-state system, and take the form
with and real, positive constants and .
(a) Consider first the case and . Use the results of degenerate perturbation theory to obtain the energy eigenvalues correct to order .
(b) Now consider the different case and . Use the results of non-degenerate perturbation theory to obtain the energy eigenvalues correct to order . Why is it not necessary to use degenerate perturbation theory in this case?
(c) Obtain the exact energy eigenvalues in case (b), and compare these to your perturbative results by expanding to second order in .
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Paper 2, Section II, B
2019 comment(a) Let and be two eigenstates of a time-independent Hamiltonian , separated in energy by . At time the system is perturbed by a small, time independent operator . The perturbation is turned off at time . Show that if the system is initially in state , the probability of a transition to state is approximately
(b) An uncharged particle with spin one-half and magnetic moment travels at speed through a region of uniform magnetic field . Over a length of its path, an additional perpendicular magnetic field is applied. The spin-dependent part of the Hamiltonian is
where and are Pauli matrices. The particle initially has its spin aligned along the direction of . Find the probability that it makes a transition to the state with opposite spin
(i) by assuming and using your result from part (a),
(ii) by finding the exact evolution of the state.
[Hint: for any 3-vector , where is the unit matrix, and
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Paper 1, Section II, B
2019 commentA isotropic harmonic oscillator of mass and frequency has lowering operators
where and are the position and momentum operators. Assuming the standard commutation relations for and , evaluate the commutators and , for , among the components of the raising and lowering operators.
How is the ground state of the oscillator defined? How are normalised higher excited states obtained from ? [You should determine the appropriate normalisation constant for each energy eigenstate.]
By expressing the orbital angular momentum operator in terms of the raising and lowering operators, show that each first excited state of the isotropic oscillator has total orbital angular momentum quantum number , and find a linear combination of these first excited states obeying and .
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Paper 4, Section II, J
2019 commentWe consider a statistical model .
(a) Define the maximum likelihood estimator (MLE) and the Fisher information
(b) Let and assume there exist a continuous one-to-one function and a real-valued function such that
(i) For i.i.d. from the model for some , give the limit in almost sure sense of
Give a consistent estimator of in terms of .
(ii) Assume further that and that is continuously differentiable and strictly monotone. What is the limit in distribution of . Assume too that the statistical model satisfies the usual regularity assumptions. Do you necessarily expect for all ? Why?
(iii) Propose an alternative estimator for with smaller bias than if for some with .
(iv) Further to all the assumptions in iii), assume that the MLE for is of the form
What is the link between the Fisher information at and the variance of ? What does this mean in terms of the precision of the estimator and why?
[You may use results from the course, provided you state them clearly.]
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Paper 3, Section II, J
2019 commentWe consider the exponential model , where
We observe an i.i.d. sample from the model.
(a) Compute the maximum likelihood estimator for . What is the limit in distribution of ?
(b) Consider the Bayesian setting and place a , prior for with density
where is the Gamma function satisfying for all . What is the posterior distribution for ? What is the Bayes estimator for the squared loss?
(c) Show that the Bayes estimator is consistent. What is the limiting distribution of ?
[You may use results from the course, provided you state them clearly.]
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Paper 2, Section II, J
2019 comment(a) We consider the model and an i.i.d. sample from it. Compute the expectation and variance of and check they are equal. Find the maximum likelihood estimator for and, using its form, derive the limit in distribution of .
(b) In practice, Poisson-looking data show overdispersion, i.e., the sample variance is larger than the sample expectation. For and , let ,
Show that this defines a distribution. Does it model overdispersion? Justify your answer.
(c) Let be an i.i.d. sample from . Assume is known. Find the maximum likelihood estimator for .
(d) Furthermore, assume that, for any converges in distribution to a random variable as . Suppose we wanted to test the null hypothesis that our data arises from the model in part (a). Before making any further computations, can we necessarily expect to follow a normal distribution under the null hypothesis? Explain. Check your answer by computing the appropriate distribution.
[You may use results from the course, provided you state it clearly.]
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Paper 1, Section II, J
2019 commentIn a regression problem, for a given fixed, we observe such that
for an unknown and random such that for some known .
(a) When and has rank , compute the maximum likelihood estimator for . When , what issue is there with the likelihood maximisation approach and how many maximisers of the likelihood are there (if any)?
(b) For any fixed, we consider minimising
over . Derive an expression for and show it is well defined, i.e., there is a unique minimiser for every and .
Assume and that has rank . Let and note that for some orthogonal matrix and some diagonal matrix whose diagonal entries satisfy . Assume that the columns of have mean zero.
(c) Denote the columns of by . Show that they are sample principal components, i.e., that their pairwise sample correlations are zero and that they have sample variances , respectively. [Hint: the sample covariance between and is .]
(d) Show that
Conclude that prediction is the closest point to within the subspace spanned by the normalised sample principal components of part (c).
(e) Show that
Assume for some . Conclude that prediction is approximately the closest point to within the subspace spanned by the normalised sample principal components of part (c) with the greatest variance.
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Paper 2, Section II, K
2019 comment(a) Let for be two measurable spaces. Define the product -algebra on the Cartesian product . Given a probability measure on for each , define the product measure . Assuming the existence of a product measure, explain why it is unique. [You may use standard results from the course if clearly stated.]
(b) Let be a probability space on which the real random variables and are defined. Explain what is meant when one says that has law . On what measurable space is the measure defined? Explain what it means for and to be independent random variables.
(c) Now let , let be its Borel -algebra and let be Lebesgue measure. Give an example of a measure on the product such that for every Borel set , but such that is not Lebesgue measure on .
(d) Let be as in part (c) and let be intervals of length and respectively. Show that
(e) Let be as in part (c). Fix and let denote the projection from to . Construct a probability measure on , such that the image under each coincides with the -dimensional Lebesgue measure, while itself is not the -dimensional Lebesgue measure. Hint: Consider the following collection of independent random variables: uniformly distributed on , and such that for each
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Paper 3, Section II, K
2019 comment(a) Let and be real random variables such that for every compactly supported continuous function . Show that and have the same law.
(b) Given a real random variable , let be its characteristic function. Prove the identity
for real , where is is continuous and compactly supported, and where is a Lebesgue integrable function such that is also Lebesgue integrable, where
is its Fourier transform. Use the above identity to derive a formula for in terms of , and recover the fact that determines the law of uniquely.
(c) Let and be bounded random variables such that for every positive integer . Show that and have the same law.
(d) The Laplace transform of a non-negative random variable is defined by the formula
for . Let and be (possibly unbounded) non-negative random variables such that for all . Show that and have the same law.
(e) Let
where is a non-negative integer and is the indicator function of the interval .
Given non-negative integers , suppose that the random variables are independent with having density function . Find the density of the random variable .
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Paper 4, Section II, K
2019 comment(a) Let and be real random variables with finite second moment on a probability space . Assume that converges to almost surely. Show that the following assertions are equivalent:
(i) in as
(ii) as .
(b) Suppose now that is the Borel -algebra of and is Lebesgue measure. Given a Borel probability measure on we set
where is the distribution function of and .
(i) Show that is a random variable on with law .
(ii) Let and be Borel probability measures on with finite second moments. Show that
if and only if converges weakly to and converges to as
[You may use any theorem proven in lectures as long as it is clearly stated. Furthermore, you may use without proof the fact that converges weakly to as if and only if converges to almost surely.]
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Paper 1, Section II, K
2019 commentLet be an -valued random variable. Given we let
be its characteristic function, where is the usual inner product on .
(a) Suppose is a Gaussian vector with mean 0 and covariance matrix , where and is the identity matrix. What is the formula for the characteristic function in the case ? Derive from it a formula for in the case .
(b) We now no longer assume that is necessarily a Gaussian vector. Instead we assume that the 's are independent random variables and that the random vector has the same law as for every orthogonal matrix . Furthermore we assume that .
(i) Show that there exists a continuous function such that
[You may use the fact that for every two vectors such that there is an orthogonal matrix such that . ]
(ii) Show that for all
(iii) Deduce that takes values in , and furthermore that there exists such that , for all .
(iv) What must be the law of ?
[Standard properties of characteristic functions from the course may be used without proof if clearly stated.]
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Paper 4, Section I,
2019 comment(a) Define the order of for coprime integers and with . Explain briefly how knowledge of this order can be used to provide a factor of , stating conditions on and its order that must be satisfied.
(b) Shor's algorithm for factoring starts by choosing coprime. Describe the subsequent steps of a single run of Shor's algorithm that computes the order of mod with probability .
[Any significant theorems that you invoke to justify the algorithm should be clearly stated (but proofs are not required). In addition you may use without proof the following two technical results.
Theorem : For positive integers and with , and any , let be the largest integer such that Let QFT denote the quantum Fourier transform . Suppose we measure to obtain an integer with Then with probability will be an integer closest to a multiple of for which the value of (between 0 and ) is coprime to .
Theorem CF: For any rational number with and with integers a and having at most digits each, let with and coprime, be any rational number satisfying
Then is one of the convergents of the continued fraction of and all the convergents can be classically computed from in time .]
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Paper 3, Section I,
2019 commentLet denote the set of all -bit strings and write . Let denote the identity operator on qubits and for introduce the -qubit operator
where is the Hadamard operation on each of the qubits, and and are given by
Also introduce the states
Let denote the real span of and .
(a) Show that maps to itself, and derive a geometrical interpretation of the action of on , stating clearly any results from Euclidean geometry that you use.
(b) Let be the Boolean function such that iff . Suppose that . Show that we can obtain an with certainty by using just one application of the standard quantum oracle for (together with other operations that are independent of ).
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Paper 2, Section I,
2019 commentThe BB84 quantum key distribution protocol begins with Alice choosing two uniformly random bit strings and .
(a) In terms of these strings, describe Alice's process of conjugate coding for the BB84 protocol.
(b) Suppose Alice and Bob are distantly separated in space and have available a noiseless quantum channel on which there is no eavesdropping. They can also communicate classically publicly. For this idealised situation, describe the steps of the BB84 protocol that results in Alice and Bob sharing a secret key of expected length .
(c) Suppose now that an eavesdropper Eve taps into the channel and carries out the following action on each passing qubit. With probability , Eve lets it pass undisturbed, and with probability she chooses a bit uniformly at random and measures the qubit in basis where and . After measurement Eve sends the post-measurement state on to Bob. Calculate the bit error rate for Alice and Bob's final key in part (b) that results from Eve's action.
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Paper 1, Section I, Introduce the 2 -qubit states
2019 commentwhere and are the standard qubit Pauli operations and .
(a) For any 1-qubit state show that the 3 -qubit state of system can be expressed as
where the 1 -qubit states are uniquely determined. Show that .
(b) In addition to you may now assume that . Alice and Bob are separated distantly in space and share a state with and labelling qubits held by Alice and Bob respectively. Alice also has a qubit in state whose identity is unknown to her. Using the results of part (a) show how she can transfer the state of to Bob using only local operations and classical communication, i.e. the sending of quantum states across space is not allowed.
(c) Suppose that in part (b), while sharing the state, Alice and Bob are also unable to engage in any classical communication, i.e. they are able only to perform local operations. Can Alice now, perhaps by a modified process, transfer the state of to Bob? Give a reason for your answer.
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Paper 3, Section II, D
2019 commentLet denote a -dimensional state space with orthonormal basis . For any let be the operator on defined by
for all and .
(a) Define , the quantum Fourier transform (for any chosen .
(b) Let on (for any chosen ) denote the operator defined by
for . Show that the Fourier basis states for are eigenstates of . By expressing in terms of find a basis of eigenstates of and determine the corresponding eigenvalues.
(c) Consider the following oracle promise problem:
Input: an oracle for a function .
Promise: has the form where and are unknown coefficients (and with all arithmetic being .
Problem: Determine with certainty.
Can this problem be solved by a single query to a classical oracle for (and possible further processing independent of ? Give a reason for your answer.
Using the results of part (b) or otherwise, give a quantum algorithm for this problem that makes just one query to the quantum oracle for .
(d) For any , let and (all arithmetic being ). Show how and can each be implemented with one use of together with other unitary gates that are independent of .
(e) Consider now the oracle problem of the form in part (c) except that now is a quadratic function with unknown coefficients (and all arithmetic being mod 3), and the problem is to determine the coefficient with certainty. Using the results of part (d) or otherwise, give a quantum algorithm for this problem that makes just two queries to the quantum oracle for .
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Paper 2, Section II, D
2019 commentLet be two quantum states and let and be associated probabilities with and . Alice chooses state with probability and sends it to Bob. Upon receiving it, Bob performs a 2-outcome measurement with outcomes labelled 0 and 1 , in an attempt to identify which state Alice sent.
(a) By using the extremal property of eigenvalues, or otherwise, show that the operator has exactly two nonzero eigenvalues, one of which is positive and the other negative.
(b) Let denote the probability that Bob correctly identifies Alice's sent state. If the measurement comprises orthogonal projectors (corresponding to outcomes 0 and 1 respectively) give an expression for in terms of and .
(c) Show that the optimal success probability , i.e. the maximum attainable value of , is
where .
(d) Suppose we now place the following extra requirement on Bob's discrimination process: whenever Bob obtains output 0 then the state sent by Alice was definitely . Show that Bob's now satisfies .
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Paper 3, Section II, I
2019 commentIn this question all representations are complex and is a finite group.
(a) State and prove Mackey's theorem. State the Frobenius reciprocity theorem.
(b) Let be a finite -set and let be the corresponding permutation representation. Pick any orbit of on : it is isomorphic as a -set to for some subgroup of . Write down the character of .
(i) Let be the trivial representation of . Show that may be written as a direct sum
for some representation .
(ii) Using the results of (a) compute the character inner product in terms of the number of double cosets.
(iii) Now suppose that , so that . By writing as a direct sum of irreducible representations, deduce from (ii) that the representation is irreducible if and only if acts 2 -transitively. In that case, show that is not the trivial representation.
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Paper 4, Section II, I
2019 comment(a) What is meant by a compact topological group? Explain why is an example of such a group.
[In the following the existence of a Haar measure for any compact Hausdorff topological group may be assumed, if required.]
(b) Let be any compact Hausdorff topological group. Show that there is a continuous group homomorphism if and only if has an -dimensional representation over . [Here denotes the subgroup of preserving the standard (positive-definite) symmetric bilinear form.]
(c) Explicitly construct such a representation by showing that acts on the following vector space of matrices,
by conjugation.
Show that
(i) this subspace is isomorphic to ;
(ii) the trace map induces an invariant positive definite symmetric bilinear form;
(iii) is surjective with kernel . [You may assume, without proof, that is connected.]
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Paper 2, Section II, I
2019 comment(a) For any finite group , let be a complete set of non-isomorphic complex irreducible representations of , with dimensions , respectively. Show that
(b) Let be the matrices
and let . Write .
(i) Prove that the derived subgroup .
(ii) Show that for all , and deduce that is a 2-group of order at most 32 .
(iii) Prove that the given representation of of degree 4 is irreducible.
(iv) Prove that has order 32 , and find all the irreducible representations of .
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Paper 1, Section II, I
2019 comment(a) State and prove Schur's lemma over .
In the remainder of this question we work over .
(b) Let be the cyclic group of order 3 .
(i) Write the regular -module as a direct sum of irreducible submodules.
(ii) Find all the intertwining homomorphisms between the irreducible -modules. Deduce that the conclusion of Schur's lemma is false if we replace by .
(c) Henceforth let be a cyclic group of order . Show that
(i) if is even, the regular -module is a direct sum of two (non-isomorphic) 1dimensional irreducible submodules and (non-isomorphic) 2-dimensional irreducible submodules;
(ii) if is odd, the regular -module is a direct sum of one 1-dimensional irreducible submodule and (non-isomorphic) 2-dimensional irreducible submodules.
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Paper 3, Section II, F
2019 commentLet be a lattice in , and a holomorphic map of complex tori. Show that lifts to a linear map .
Give the definition of , the Weierstrass -function for . Show that there exist constants such that
Suppose , that is, is a biholomorphic group homomorphism. Prove that there exists a lift of , where is a root of unity for which there exist such that .
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Paper 2, Section II, F
2019 comment(a) Prove that as a map from the upper half-plane to is a covering map which is not regular.
(b) Determine the set of singular points on the unit circle for
(c) Suppose is a holomorphic map where is the unit disk. Prove that extends to a holomorphic map . If additionally is biholomorphic, prove that .
(d) Suppose that is a holomorphic injection with a compact Riemann surface. Prove that has genus 0 , stating carefully any theorems you use.
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Paper 1, Section II, F
2019 commentDefine .
(a) Prove by defining an atlas that is a Riemann surface.
(b) Now assume that by adding finitely many points, it is possible to compactify to a Riemann surface so that the coordinate projections extend to holomorphic maps and from to . Compute the genus of .
(c) Assume that any holomorphic automorphism of extends to a holomorphic automorphism of . Prove that the group Aut of holomorphic automorphisms of contains an element of order 7 . Prove further that there exists a holomorphic map which satisfies .
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Paper 4, Section I, J
2019 commentIn a normal linear model with design matrix , output variables and parameters and , define a -level prediction interval for a new observation with input variables . Derive an explicit formula for the interval, proving that it satisfies the properties required by the definition. [You may assume that the maximum likelihood estimator is independent of , which has a distribution.]
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Paper 3, Section I, J
2019 comment(a) For a given model with likelihood , define the Fisher information matrix in terms of the Hessian of the log-likelihood.
Consider a generalised linear model with design matrix , output variables , a bijective link function, mean parameters and dispersion parameters . Assume is known.
(b) State the form of the log-likelihood.
(c) For the canonical link, show that when the parameter is known, the Fisher information matrix is equal to
for a diagonal matrix depending on the means . Identify .
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Paper 2, Section I, J
2019 commentThe cycling data frame contains the results of a study on the effects of cycling to work among 1,000 participants with asthma, a respiratory illness. Half of the participants, chosen uniformly at random, received a monetary incentive to cycle to work, and the other half did not. The variables in the data frame are:
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miles: the average number of miles cycled per week
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episodes: the number of asthma episodes experienced during the study
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incentive: whether or not a monetary incentive to cycle was given
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history: the number of asthma episodes in the year preceding the study
Consider the code below and its abbreviated output.
(episodes miles history, data=cycling)
Coefficients:
Estimate Std. Error value
(Intercept)
miles
history
episodes incentive history, data=cycling)
summary (lm.2)
Coefficients:
Estimate Std. Error value
(Intercept)
incentiveYes
history
miles incentive history, data=cycling)
Coefficients :
Estimate Std. Error t value
(Intercept)
incentiveYes
history
(a) For each of the fitted models, briefly explain what can be inferred about participants with similar histories.
(b) Based on this analysis and the experimental design, is it advisable for a participant with asthma to cycle to work more often? Explain.
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Paper 1, Section I, J
2019 commentThe Gamma distribution with shape parameter and scale parameter has probability density function
Give the definition of an exponential dispersion family and show that the set of Gamma distributions forms one such family. Find the cumulant generating function and derive the mean and variance of the Gamma distribution as a function of and .
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Paper 4, Section II, J
2019 commentA sociologist collects a dataset on friendships among Cambridge graduates. Let if persons and are friends 3 years after graduation, and otherwise. Let be a categorical variable for person 's college, taking values in the set . Consider logistic regression models,
with parameters either
-
; or,
-
; or,
-
, where if and 0 otherwise.
(a) Write the likelihood of the models.
(b) Show that the three models are nested and specify the order. Suggest a statistic to compare models 1 and 3, give its definition and specify its asymptotic distribution under the null hypothesis, citing any necessary theorems.
(c) Suppose persons and are in the same college consider the number of friendships, and , that each of them has with people in college ( and fixed). In each of the models above, compare the distribution of these two random variables. Explain why this might lead to a poor quality of fit.
(d) Find a minimal sufficient statistic for model 3. [You may use the following characterisation of a minimal sufficient statistic: let be the likelihood in this model, where and suppose is a statistic such that is constant in if and only if ; then, is a minimal sufficient statistic for .]
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Paper 1, Section II, J
2019 commentThe ice_cream data frame contains the result of a blind tasting of 90 ice creams, each of which is rated as poor, good, or excellent. It also contains the price of each ice cream classified into three categories. Consider the code below and its output.

(a) Write down the generalised linear model fitted by the code above.
(b) Prove that the fitted values resulting from the maximum likelihood estimator of the coefficients in this model are identical to those resulting from the maximum likelihood estimator when fitting a Multinomial model which assumes the number of ice creams at each price level is fixed.
(c) Using the output above, perform a goodness-of-fit test at the level, specifying the null hypothesis, the test statistic, its asymptotic null distribution, any assumptions of the test and the decision from your test. (d) If we believe that better ice creams are more expensive, what could be a more powerful test against the model fitted above and why?
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Paper 4, Section II, D
2019 commentGive an outline of the Landau theory of phase transitions for a system with one real order parameter . Describe the phase transitions that can be modelled by the Landau potentials (i) , (ii) ,
where and are control parameters that depend on the temperature and pressure.
In case (ii), find the curve of first-order phase transitions in the plane. Find the region where it is possible for superheating to occur. Find also the region where it is possible for supercooling to occur.
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Paper 3, Section II, D
2019 commentWhat is meant by the chemical potential of a thermodynamic system? Derive the Gibbs distribution for a system at temperature and chemical potential (and fixed volume) with variable particle number .
Consider a non-interacting, two-dimensional gas of fermionic particles in a region of fixed area, at temperature and chemical potential . Using the Gibbs distribution, find the mean occupation number of a one-particle quantum state of energy . Show that the density of states is independent of and deduce that the mean number of particles between energies and is very well approximated for by
where is the Fermi energy. Show that, for small, the heat capacity of the gas has a power-law dependence on , and find the power.
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Paper 2, Section II, D
2019 commentUsing the classical statistical mechanics of a gas of molecules with negligible interactions, derive the deal gas law. Explain briefly to what extent this law is independent of the molecule's internal structure.
Calculate the entropy of a monatomic gas of low density, with negligible interactions. Deduce the equation relating the pressure and volume of the gas on a curve in the -plane along which is constant.
[You may use for
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Paper 1, Section II, D
2019 comment(a) Explain, from a macroscopic and microscopic point of view, what is meant by an adiabatic change. A system has access to heat baths at temperatures and , with . Show that the most effective method for repeatedly converting heat to work, using this system, is by combining isothermal and adiabatic changes. Define the efficiency and calculate it in terms of and .
(b) A thermal system (of constant volume) undergoes a phase transition at temperature . The heat capacity of the system is measured to be
where are constants. A theoretical calculation of the entropy for leads to
How can the value of the theoretically-obtained constant be verified using macroscopically measurable quantities?
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Paper 4, Section II, K
2019 comment(a) Describe the (Cox-Ross-Rubinstein) binomial model. What are the necessary and sufficient conditions on the model parameters for it to be arbitrage-free? How is the equivalent martingale measure characterised in this case?
(b) Consider a discounted claim of the form for some function . Show that the value process of is of the form
for , where the function is given by
You may use any property of conditional expectations without proof.
(c) Suppose that only depends on the terminal value of the stock price. Derive an explicit formula for the value of at time .
(d) Suppose that is of the form , where . Show that the value process of is of the form
for , where the function is given by
for a function to be determined.
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Paper 3, Section II, K
2019 commentIn the Black-Scholes model the price at time 0 for a European option of the form with maturity is given by
(a) Find the price at time 0 of a European call option with maturity and strike price in terms of the standard normal distribution function. Derive the put-call parity to find the price of the corresponding European put option.
(b) The digital call option with maturity and strike price has payoff given by
What is the value of the option at any time ? Determine the number of units of the risky asset that are held in the hedging strategy at time .
(c) The digital put option with maturity and strike price has payoff
Find the put-call parity for digital options and deduce the Black-Scholes price at time 0 for a digital put.
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Paper 2, Section II,
2019 comment(a) In the context of a multi-period model in discrete time, what does it mean to say that a probability measure is an equivalent martingale measure?
(b) State the fundamental theorem of asset pricing.
(c) Consider a single-period model with one risky asset having initial price . At time 1 its value is a random variable on of the form
where . Assume that there is a riskless numéraire with . Show that there is no arbitrage in this model.
[Hint: You may find it useful to consider a density of the form and find suitable and . You may use without proof that if is a normal random variable then .]
(d) Now consider a multi-period model with one risky asset having a non-random initial price and a price process of the form
where are i.i.d. -distributed random variables on . Assume that there is a constant riskless numéraire with for all . Show that there exists no arbitrage in this model.
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Paper 1, Section II, 30K
2019 comment(a) What does it mean to say that is a martingale? (b) Let be a Markov chain defined by and
and
for . Show that is a martingale with respect to the filtration where is trivial and for .
(c) Let be adapted with respect to a filtration with for all . Show that the following are equivalent:
(i) is a martingale.
(ii) For every stopping time , the stopped process defined by , , is a martingale.
(iii) for all and every stopping time .
[Hint: To show that (iii) implies (i) you might find it useful to consider the stopping time
for any
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Paper 4, Section I, H
2019 commentShow that is irrational. [Hint: consider the functions given by
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Paper 3, Section I, H
2019 commentState Nash's theorem for a non zero-sum game in the case of two players with two choices.
The role playing game Tixerb involves two players. Before the game begins, each player chooses a with which they announce. They may change their choice as many times as they wish, but, once the game begins, no further changes are allowed. When the game starts, player becomes a Dark Lord with probability and a harmless peasant with probability . If one player is a Dark Lord and the other a peasant the Lord gets 2 points and the peasant . If both are peasants they get 1 point each, if both Lords they get each. Show that there exists a , to be found, such that, if there will be three choices of for which neither player can increase the expected value of their outcome by changing their choice unilaterally, but, if , there will only be one. Find the appropriate in each case.
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Paper 2, Section I, H
2019 commentLet be the collection of non-empty closed bounded subsets of .
(a) Show that, if and we write
then .
(b) Show that, if , and
then .
(c) Assuming the result that
defines a metric on (the Hausdorff metric), show that if and are as in part (b), then as .
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Paper 1, Section I, H
2019 commentLet be the th Chebychev polynomial. Suppose that for all and that converges. Explain why is a well defined continuous function on .
Show that, if we take , we can find points with
such that for each .
Suppose that is a decreasing sequence of positive numbers and that as . Stating clearly any theorem that you use, show that there exists a continuous function with
for all polynomials of degree at most and all .
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Paper 2, Section II, H
2019 commentThroughout this question denotes the closed interval .
(a) For , consider the points with and . Show that, if we colour them red or green in such a way that and 1 are coloured differently, there must be two neighbouring points of different colours.
(b) Deduce from part (a) that, if with and closed, and , then .
(c) Deduce from part (b) that there does not exist a continuous function with for all and .
(d) Deduce from part (c) that if is continuous then there exists an with .
(e) Deduce the conclusion of part (c) from the conclusion of part (d).
(f) Deduce the conclusion of part (b) from the conclusion of part (c).
(g) Suppose that we replace wherever it occurs by the unit circle
Which of the conclusions of parts (b), (c) and (d) remain true? Give reasons.
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Paper 4, Section II, H
2019 comment(a) Suppose that is a non-empty subset of the square and is analytic in the larger square for some . Show that can be uniformly approximated on by polynomials.
(b) Let be a closed non-empty proper subset of . Let be the set of such that can be approximated uniformly on by polynomials and let . Show that and are open. Is it always true that is non-empty? Is it always true that, if is bounded, then is empty? Give reasons.
[No form of Runge's theorem may be used without proof.]
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Paper 4, Section II, A
2019 comment(a) Assuming a slowly-varying two-dimensional wave pattern of the form
where , and a local dispersion relation , derive the ray tracing equations,
for , explaining carefully the meaning of the notation used.
(b) For a homogeneous, time-independent (but not necessarily isotropic) medium, show that all rays are straight lines. When the waves have zero frequency, deduce that if the point lies on a ray emanating from the origin in the direction given by a unit vector , then
(c) Consider a stationary obstacle in a steadily moving homogeneous medium which has the dispersion relation
where is the velocity of the medium and is a constant. The obstacle generates a steady wave system. Writing , with , show that the wave satisfies
where is defined by
with and . Deduce that the wave pattern occupies a wedge of semi-angle , extending in the negative -direction.
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Paper 2, Section II, A
2019 commentThe linearised equation of motion governing small disturbances in a homogeneous elastic medium of density is
where is the displacement, and and are the Lamé moduli.
(a) The medium occupies the region between a rigid plane boundary at and a free surface at . Show that waves can propagate in the -direction within this region, and find the dispersion relation for such waves.
(b) For each mode, deduce the cutoff frequency, the phase velocity and the group velocity. Plot the latter two velocities as a function of wavenumber.
(c) Verify that in an average sense (to be made precise), the wave energy flux is equal to the wave energy density multiplied by the group velocity.
[You may assume that the elastic energy per unit volume is given by
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Paper 3, Section II, A
2019 comment(a) Derive the wave equation for perturbation pressure for linearised sound waves in a compressible gas.
(b) For a single plane wave show that the perturbation pressure and the velocity are linearly proportional and find the constant of proportionality, i.e. the acoustic impedance.
(c) Gas occupies a tube lying parallel to the -axis. In the regions and the gas has uniform density and sound speed . For the temperature of the gas has been adjusted so that it has uniform density and sound speed . A harmonic plane wave with frequency and unit amplitude is incident from . If is the (in general complex) amplitude of the wave transmitted into , show that
where and . Discuss both of the limits and .
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Paper 1, Section II, A
2019 commentThe equation of state relating pressure to density for a perfect gas is given by
where and are constants, and is the specific heat ratio.
(a) Starting from the equations for one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants,
are constant on characteristics given by
where is the velocity of the gas, is the local speed of sound, and is a constant.
(b) Such an ideal gas initially occupies the region to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest. At time the piston starts moving to the left with path given by
(i) Solve for and in the region under the assumptions that and that is monotonically increasing, where dot indicates a time derivative.
[It is sufficient to leave the solution in implicit form, i.e. for given you should not attempt to solve the characteristic equation explicitly.]
(ii) Briefly outline the behaviour of and for times , where is the solution to .
(iii) Now suppose,
where . For , find a leading-order approximation to the solution of the characteristic equation when and .
[Hint: You may find it useful to consider the structure of the characteristics in the limiting case when .]